#set document(title: "3.5 Joint Variation", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 3.5#h(0.6em)Joint Variation === Functions of Two or More Variables So far, we have studied functions that relate values of an output variable to values of a single input variable. But it is not uncommon for an output variable to depend on two or more inputs. Many familiar formulas describe functions of several variables. For example, the perimeter of a rectangle depends on its length and width. The volume of a cylinder depends on its radius and height. The distance you travel depends on your speed and the time you spent traveling. Each of these formulas can be written with function notation. #math.equation(block: true, alt: "P equals f open parenthesis l , w close parenthesis equals 2 l plus 2 w, Perimeter is a function of length and width.; V equals f open parenthesis r , h close parenthesis equals π r squared h, Volume is a function of radius and height.; d equals f open parenthesis r , t close parenthesis equals r t, Distance is a function of rate and time.")[$& P = f ( l , w ) = 2 l + 2 w & & "Perimeter is a function of length and width." \ & V = f ( r , h ) = π r^(2) h & & "Volume is a function of radius and height." \ & d = f ( r , t ) = r t & & "Distance is a function of rate and time."$] #examplebox("Example 1")[][ The cost, #math.equation(block: false, alt: "C")[$C$], of driving a rental car is given by the function #math.equation(block: true, alt: "C equals f open parenthesis t , g close parenthesis equals 129.95 t plus 3.80 g")[$C = f ( t , g ) = 129.95 t + 3.80 g$] where #math.equation(block: false, alt: "t")[$t$] is the number of days you rent the car and #math.equation(block: false, alt: "g")[$g$] is the number of gallons of gas you buy. + Evaluate #math.equation(block: false, alt: "f open parenthesis 3 , 10 close parenthesis")[$f ( 3 , 10 )$] and explain what it means. + You have \$350 to rent a car for #math.equation(block: false, alt: "2")[$2$] days. How much gas can you buy? #solutionbox[ + We substitute #math.equation(block: false, alt: "3")[$3$] for #math.equation(block: false, alt: "t")[$t$] and #math.equation(block: false, alt: "10")[$10$] for #math.equation(block: false, alt: "g")[$g$] to find #math.equation(block: true, alt: "f open parenthesis 3 , 10 close parenthesis equals 129.95 open parenthesis 3 close parenthesis plus 3.80 open parenthesis 10 close parenthesis equals 417.85")[$f ( 3 , 10 ) = 129.95 ( 3 ) + 3.80 ( 10 ) = 417.85$] It will cost \$#math.equation(block: false, alt: "417.85")[$417.85$] to rent a car for #math.equation(block: false, alt: "3")[$3$] days and buy #math.equation(block: false, alt: "10")[$10$] gallons of gas. + We would like to find the value of #math.equation(block: false, alt: "g")[$g$] when #math.equation(block: false, alt: "t equals 2")[$t = 2$] and #math.equation(block: false, alt: "C equals 350")[$C = 350$]. That is, we want to solve the equation #math.equation(block: true, alt: "350, equals f open parenthesis 2 , g close parenthesis equals 129.95 open parenthesis 2 close parenthesis plus 3.80 g; 350, equals 259.90 plus 3.80 g; g, equals 23.7")[$350 & = f ( 2 , g ) = 129.95 ( 2 ) + 3.80 g \ 350 & = 259.90 + 3.80 g \ g & = 23.7$] You can buy #math.equation(block: false, alt: "23.7")[$23.7$] gallons of gas. ] ] Which is true about a function of two variables? \_\_\_\_\_ #solutionbox[ A function of two variables has two inputs and one output. ] Which statement is true about a function of two variables? + It has one input and two outputs. + It has two inputs and one output. + It has two inputs and two outputs. + None of the above. The maximum height that the water stream from a fire hose can reach depends on the water pressure and the diameter of the nozzle, and is given by the function #math.equation(block: true, alt: "H equals f open parenthesis P , n close parenthesis equals 26 plus the fraction 5 over 8 P plus 5 n")[$H = f ( P , n ) = 26 + display(frac(5, 8)) P + 5 n$] where #math.equation(block: false, alt: "P")[$P$] is the nozzle pressure in psi, and #math.equation(block: false, alt: "n")[$n$] is measured in #math.equation(block: false, alt: "the fraction 1 over 8")[$display(frac(1, 8))$]-inch increments over the standard nozzle diameter of #math.equation(block: false, alt: "the fraction 3 over 4")[$display(frac(3, 4))$] inch. + Evaluate #math.equation(block: false, alt: "f open parenthesis 40 , 2 close parenthesis")[$f ( 40 , 2 )$] and explain what it means. #linebreak() #math.equation(block: false, alt: "f open parenthesis 40 , 2 close parenthesis equals")[$f ( 40 , 2 ) =$]\_\_\_\_\_ #linebreak() \_\_\_\_\_ + What nozzle pressure is needed to reach a height of #math.equation(block: false, alt: "91")[$91$] feet with a #math.equation(block: false, alt: "1 the fraction 1 over 8")[$1 display(frac(1, 8))$]-inch nozzle? #linebreak() \_\_\_\_\_\_\_\_\_\_ #solutionbox[ + #math.equation(block: false, alt: "f open parenthesis 40 , 2 close parenthesis equals 61")[$f ( 40 , 2 ) = 61$]. With nozzle diameter #math.equation(block: false, alt: "1")[$1$] inch and nozzle pressure #math.equation(block: false, alt: "40")[$40$] psi, the water will reach #math.equation(block: false, alt: "61")[$61$] feet. + The #math.equation(block: false, alt: "1 the fraction 1 over 8")[$1 frac(1, 8)$]-inch nozzle is three #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$] increments over the standard nozzle diameter of #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] inch. We solve for #math.equation(block: false, alt: "P")[$P$] when #math.equation(block: false, alt: "H equals 91")[$H = 91$] and #math.equation(block: false, alt: "n equals 3")[$n = 3$]: #math.equation(block: false, alt: "P equals 80")[$P = 80$] psi ] The maximum height that the water stream from a fire hose can reach depends on the water pressure and the diameter of the nozzle, and is given by the function #math.equation(block: true, alt: "H equals f open parenthesis P , n close parenthesis equals 26 plus the fraction 5 over 8 P plus 5 n")[$H = f ( P , n ) = 26 + display(frac(5, 8)) P + 5 n$] where #math.equation(block: false, alt: "P")[$P$] is the nozzle pressure in psi, and #math.equation(block: false, alt: "n")[$n$] is measured in #math.equation(block: false, alt: "the fraction 1 over 8")[$display(frac(1, 8))$]-inch increments over the standard nozzle diameter of #math.equation(block: false, alt: "the fraction 3 over 4")[$display(frac(3, 4))$] inch. + Evaluate #math.equation(block: false, alt: "f open parenthesis 40 , 2 close parenthesis")[$f ( 40 , 2 )$] and explain what it means. + What nozzle pressure is needed to reach a height of #math.equation(block: false, alt: "91")[$91$] feet with a #math.equation(block: false, alt: "1 the fraction 1 over 8")[$1 display(frac(1, 8))$]-inch nozzle? #solutionbox[ + #math.equation(block: false, alt: "f open parenthesis 40 , 2 close parenthesis equals 61")[$f ( 40 , 2 ) = 61$]. With nozzle diameter #math.equation(block: false, alt: "1")[$1$] inch and nozzle pressure #math.equation(block: false, alt: "40")[$40$] psi, the water will reach #math.equation(block: false, alt: "61")[$61$] feet. + The #math.equation(block: false, alt: "1 the fraction 1 over 8")[$1 frac(1, 8)$]-inch nozzle is three #math.equation(block: false, alt: "the fraction 1 over 8")[$frac(1, 8)$] increments over the standard nozzle diameter of #math.equation(block: false, alt: "the fraction 3 over 4")[$frac(3, 4)$] inch. We solve for #math.equation(block: false, alt: "P")[$P$] when #math.equation(block: false, alt: "H equals 91")[$H = 91$] and #math.equation(block: false, alt: "n equals 3")[$n = 3$]: #math.equation(block: false, alt: "P equals 80")[$P = 80$] psi ] === Tables of Values Just as for functions of a single variable, we can use tables to describe functions of two variables, #math.equation(block: false, alt: "z equals f open parenthesis x , y close parenthesis")[$z = f ( x , y )$]. The row and column headings show the values of the two input variables, and the table entries show the values of the output variable. #examplebox("Example 2")[][ Windchill is a function of two variables, temperature and wind speed, or #math.equation(block: false, alt: "W equals f open parenthesis s , t close parenthesis")[$W = f ( s , t )$]. The table shows the windchill factor for various combinations of temperature and wind speed. #figure(table( columns: 9, align: left, inset: 6pt, [Windchill Factors], [], [], [], [], [], [], [], [], [], [Temperature (#math.equation(block: false, alt: "∘")[$∘$]F)], [], [], [], [], [], [], [], [Wind speed (mph)], [#math.equation(block: false, alt: "35")[$35$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "15")[$15$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "33")[$33$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "21")[$21$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "12")[$12$]], [#math.equation(block: false, alt: "7")[$7$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "minus 5")[$− 5$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "22")[$22$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 9")[$− 9$]], [#math.equation(block: false, alt: "minus 15")[$− 15$]], [#math.equation(block: false, alt: "minus 22")[$− 22$]], [#math.equation(block: false, alt: "15")[$15$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "minus 5")[$− 5$]], [#math.equation(block: false, alt: "minus 11")[$− 11$]], [#math.equation(block: false, alt: "minus 18")[$− 18$]], [#math.equation(block: false, alt: "minus 25")[$− 25$]], [#math.equation(block: false, alt: "minus 31")[$− 31$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "12")[$12$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "minus 3")[$− 3$]], [#math.equation(block: false, alt: "minus 10")[$− 10$]], [#math.equation(block: false, alt: "minus 17")[$− 17$]], [#math.equation(block: false, alt: "minus 24")[$− 24$]], [#math.equation(block: false, alt: "minus 31")[$− 31$]], [#math.equation(block: false, alt: "minus 39")[$− 39$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "minus 7")[$− 7$]], [#math.equation(block: false, alt: "minus 15")[$− 15$]], [#math.equation(block: false, alt: "minus 22")[$− 22$]], [#math.equation(block: false, alt: "minus 29")[$− 29$]], [#math.equation(block: false, alt: "minus 36")[$− 36$]], [#math.equation(block: false, alt: "minus 44")[$− 44$]], )) + What is the windchill factor when the temperature is #math.equation(block: false, alt: "15 degrees")[$15^(∘)$]F and the wind is blowing at #math.equation(block: false, alt: "20")[$20$] mph? Write this fact with function notation. + Find a value for #math.equation(block: false, alt: "t")[$t$] so that #math.equation(block: false, alt: "f open parenthesis 10 , t close parenthesis equals minus 15")[$" " f ( 10 , t ) = − 15$]. What does this equation tell you about the windchill factor? + Solve the equation #math.equation(block: false, alt: "f open parenthesis s , 30 close parenthesis equals 1")[$" " f ( s , 30 ) = 1$]. What does this tell you about the windchill factor? #solutionbox[ + We look in the row for #math.equation(block: false, alt: "20")[$20$] mph and the column for #math.equation(block: false, alt: "15 degrees")[$15^(∘)$]. The associated windchill factor is #math.equation(block: false, alt: "minus 17")[$− 17$], so #math.equation(block: false, alt: "f open parenthesis 20 , 15 close parenthesis equals minus 17")[$f ( 20 , 15 ) = − 17$]. + We look in the row for #math.equation(block: false, alt: "s equals 10")[$s = 10$] until we find the windchill factor of #math.equation(block: false, alt: "W equals minus 15")[$W = − 15$]. The column heading for that entry is #math.equation(block: false, alt: "5")[$5$], so #math.equation(block: false, alt: "t equals 5")[$t = 5$]. When the wind speed is #math.equation(block: false, alt: "10")[$10$] mph and the windchill factor is #math.equation(block: false, alt: "minus 15")[$− 15$], the temperature is #math.equation(block: false, alt: "5 degrees")[$5^(∘)$]F. + In the #math.equation(block: false, alt: "t equals 30 degrees")[$t = 30^(∘)$]F column, we find the windchill factor of #math.equation(block: false, alt: "1")[$1$] in the #math.equation(block: false, alt: "25")[$25$]-mph row, so #math.equation(block: false, alt: "s equals 25")[$s = 25$]. The wind speed is #math.equation(block: false, alt: "25")[$25$] mph when the temperature is #math.equation(block: false, alt: "30 degrees")[$30^(∘)$]F and the windchill factor is #math.equation(block: false, alt: "1")[$1$]. ] ] A retirement plan requires employees to put aside a fixed amount of money each year until retirement. The amount accumulated, #math.equation(block: false, alt: "A")[$A$], includes 8% annual interest on employee's annual contribution, #math.equation(block: false, alt: "c")[$c$]. #math.equation(block: false, alt: "A")[$A$] is a function of #math.equation(block: false, alt: "c")[$c$] and the number of years, #math.equation(block: false, alt: "t")[$t$], that the employee makes contributions, so #math.equation(block: false, alt: "A equals f open parenthesis c , t close parenthesis")[$A = f ( c , t )$]. #figure(table( columns: 6, align: left, inset: 6pt, [Retirement Fund Balance], [], [], [], [], [], [], [Number of years of contributions], [], [], [], [], [Annual #linebreak() contribution #linebreak()], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "500")[$500$]], [#math.equation(block: false, alt: "7243")[$7243$]], [#math.equation(block: false, alt: "22 , 881")[$22 , 881$]], [#math.equation(block: false, alt: "56 , 642")[$56 , 642$]], [#math.equation(block: false, alt: "129 , 528")[$129 , 528$]], [#math.equation(block: false, alt: "286 , 885")[$286 , 885$]], [#math.equation(block: false, alt: "1000")[$1000$]], [#math.equation(block: false, alt: "14 , 487")[$14 , 487$]], [#math.equation(block: false, alt: "45 , 762")[$45 , 762$]], [#math.equation(block: false, alt: "113 , 283")[$113 , 283$]], [#math.equation(block: false, alt: "259 , 057")[$259 , 057$]], [#math.equation(block: false, alt: "573 , 770")[$573 , 770$]], [#math.equation(block: false, alt: "1500")[$1500$]], [#math.equation(block: false, alt: "21 , 730")[$21 , 730$]], [#math.equation(block: false, alt: "68 , 643")[$68 , 643$]], [#math.equation(block: false, alt: "169 , 925")[$169 , 925$]], [#math.equation(block: false, alt: "388 , 585")[$388 , 585$]], [#math.equation(block: false, alt: "860 , 655")[$860 , 655$]], [#math.equation(block: false, alt: "2000")[$2000$]], [#math.equation(block: false, alt: "28 , 973")[$28 , 973$]], [#math.equation(block: false, alt: "91 , 524")[$91 , 524$]], [#math.equation(block: false, alt: "226 , 566")[$226 , 566$]], [#math.equation(block: false, alt: "518 , 113")[$518 , 113$]], [#math.equation(block: false, alt: "1 , 147 , 540")[$1 , 147 , 540$]], [#math.equation(block: false, alt: "2500")[$2500$]], [#math.equation(block: false, alt: "43 , 460")[$43 , 460$]], [#math.equation(block: false, alt: "137 , 286")[$137 , 286$]], [#math.equation(block: false, alt: "339 , 850")[$339 , 850$]], [#math.equation(block: false, alt: "777 , 170")[$777 , 170$]], [#math.equation(block: false, alt: "1 , 721 , 310")[$1 , 721 , 310$]], [#math.equation(block: false, alt: "3000")[$3000$]], [#math.equation(block: false, alt: "50 , 703")[$50 , 703$]], [#math.equation(block: false, alt: "160 , 167")[$160 , 167$]], [#math.equation(block: false, alt: "396 , 491")[$396 , 491$]], [#math.equation(block: false, alt: "906 , 698")[$906 , 698$]], [#math.equation(block: false, alt: "2 , 008 , 196")[$2 , 008 , 196$]], )) + How much will an employee accumulate if she contributes \$500 a year for 40 years? (Do not enter any commas. For example, enter "10000" for 10,000. #linebreak() \$\_\_\_\_\_ #linebreak() Write your answer with function notation: #linebreak() #math.equation(block: false, alt: "f open parenthesis")[$f \($]\_\_\_\_\_,\_\_\_\_\_#math.equation(block: false, alt: "close parenthesis equals")[$\) =$]\_\_\_\_\_ + How much must she contribute each year in order to accumulate \$573,770 after 50 years? \$\_\_\_\_\_ #linebreak() Write your answer with function notation. #linebreak() #math.equation(block: false, alt: "f open parenthesis")[$f \($]\_\_\_\_\_,\_\_\_\_\_#math.equation(block: false, alt: "close parenthesis equals")[$\) =$]\_\_\_\_\_ + Find a value of #math.equation(block: false, alt: "t")[$t$] that solves the equation #math.equation(block: false, alt: "137 , 286 equals f open parenthesis 2500 , t close parenthesis")[$137 , 286 = f ( 2500 , t )$]. #linebreak() #math.equation(block: false, alt: "t equals")[$t =$]\_\_\_\_\_ #linebreak() What does this equation tell you about the retirement fund? #linebreak() \_\_\_\_\_ #solutionbox[ + \$129,528, #math.equation(block: false, alt: "f open parenthesis 500 , 40 close parenthesis equals 129 , 528")[$f ( 500 , 40 ) = 129 , 528$] + \$1000, #math.equation(block: false, alt: "f open parenthesis c , 50 close parenthesis equals 573 , 770")[$f ( c , 50 ) = 573 , 770$] + 20 years. If you contribute \$2500 per year for 20 years, you will accumulate \$137,286. ] A retirement plan requires employees to put aside a fixed amount of money each year until retirement. The amount accumulated, #math.equation(block: false, alt: "A")[$A$], includes 8% annual interest on employee's annual contribution, #math.equation(block: false, alt: "c")[$c$]. #math.equation(block: false, alt: "A")[$A$] is a function of #math.equation(block: false, alt: "c")[$c$] and the number of years, #math.equation(block: false, alt: "t")[$t$], that the employee makes contributions, so #math.equation(block: false, alt: "A equals f open parenthesis c , t close parenthesis")[$" " A = f ( c , t )$]. #figure(table( columns: 6, align: left, inset: 6pt, [Retirement Fund Balance], [], [], [], [], [], [], [Number of years of contributions], [], [], [], [], [Annual #linebreak() contribution #linebreak()], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "500")[$500$]], [#math.equation(block: false, alt: "7243")[$7243$]], [#math.equation(block: false, alt: "22 , 881")[$22 , 881$]], [#math.equation(block: false, alt: "56 , 642")[$56 , 642$]], [#math.equation(block: false, alt: "129 , 528")[$129 , 528$]], [#math.equation(block: false, alt: "286 , 885")[$286 , 885$]], [#math.equation(block: false, alt: "1000")[$1000$]], [#math.equation(block: false, alt: "14 , 487")[$14 , 487$]], [#math.equation(block: false, alt: "45 , 762")[$45 , 762$]], [#math.equation(block: false, alt: "113 , 283")[$113 , 283$]], [#math.equation(block: false, alt: "259 , 057")[$259 , 057$]], [#math.equation(block: false, alt: "573 , 770")[$573 , 770$]], [#math.equation(block: false, alt: "1500")[$1500$]], [#math.equation(block: false, alt: "21 , 730")[$21 , 730$]], [#math.equation(block: false, alt: "68 , 643")[$68 , 643$]], [#math.equation(block: false, alt: "169 , 925")[$169 , 925$]], [#math.equation(block: false, alt: "388 , 585")[$388 , 585$]], [#math.equation(block: false, alt: "860 , 655")[$860 , 655$]], [#math.equation(block: false, alt: "2000")[$2000$]], [#math.equation(block: false, alt: "28 , 973")[$28 , 973$]], [#math.equation(block: false, alt: "91 , 524")[$91 , 524$]], [#math.equation(block: false, alt: "226 , 566")[$226 , 566$]], [#math.equation(block: false, alt: "518 , 113")[$518 , 113$]], [#math.equation(block: false, alt: "1 , 147 , 540")[$1 , 147 , 540$]], [#math.equation(block: false, alt: "2500")[$2500$]], [#math.equation(block: false, alt: "36 , 216")[$36 , 216$]], [#math.equation(block: false, alt: "114 , 405")[$114 , 405$]], [#math.equation(block: false, alt: "283 , 208")[$283 , 208$]], [#math.equation(block: false, alt: "647 , 641")[$647 , 641$]], [#math.equation(block: false, alt: "1 , 434 , 425")[$1 , 434 , 425$]], [#math.equation(block: false, alt: "3000")[$3000$]], [#math.equation(block: false, alt: "43 , 460")[$43 , 460$]], [#math.equation(block: false, alt: "137 , 286")[$137 , 286$]], [#math.equation(block: false, alt: "339 , 850")[$339 , 850$]], [#math.equation(block: false, alt: "777 , 170")[$777 , 170$]], [#math.equation(block: false, alt: "1 , 721 , 310")[$1 , 721 , 310$]], )) + How much will an employee accumulate if she contributes \$500 a year for 40 years? Write your answer with function notation. + How much must she contribute each year in order to accumulate \$573,770 after 50 years? Write your answer with function notation. + Find a value of #math.equation(block: false, alt: "t")[$t$] that solves the equation #math.equation(block: false, alt: "114 , 405 equals f open parenthesis 2500 , t close parenthesis")[$" " 114 , 405 = f ( 2500 , t )$]. What does this equation tell you about the retirement fund? #solutionbox[ + \$129,528, #math.equation(block: false, alt: "f open parenthesis 500 , 40 close parenthesis equals 129 , 528")[$f ( 500 , 40 ) = 129 , 528$] + \$1000, #math.equation(block: false, alt: "f open parenthesis c , 50 close parenthesis equals 573 , 770")[$f ( c , 50 ) = 573 , 770$] + 20 years. If you contribute \$2500 per year for 20 years, you will accumulate \$114,405. ] === Joint Variation Sometimes we can find patterns relating the entries in a table. #examplebox("Example 3")[][ Rectangular beams of a given length can support a load, #math.equation(block: false, alt: "L")[$L$], that depends on both the width and the depth of the beam, so that #math.equation(block: false, alt: "L equals f open parenthesis w , d close parenthesis")[$L = f ( w , d )$]. The table shows some of the values. #figure(table( columns: 7, align: left, inset: 6pt, [Maximum Load (kilograms)], [], [], [], [], [], [], [], [Depth (cm)], [], [], [], [], [], [Width (cm)], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "90")[$90$]], [#math.equation(block: false, alt: "160")[$160$]], [#math.equation(block: false, alt: "250")[$250$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "80")[$80$]], [#math.equation(block: false, alt: "180")[$180$]], [#math.equation(block: false, alt: "320")[$320$]], [#math.equation(block: false, alt: "500")[$500$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "120")[$120$]], [#math.equation(block: false, alt: "270")[$270$]], [#math.equation(block: false, alt: "480")[$480$]], [#math.equation(block: false, alt: "750")[$750$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "160")[$160$]], [#math.equation(block: false, alt: "360")[$360$]], [#math.equation(block: false, alt: "640")[$640$]], [#math.equation(block: false, alt: "1000")[$1000$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "200")[$200$]], [#math.equation(block: false, alt: "450")[$450$]], [#math.equation(block: false, alt: "800")[$800$]], [#math.equation(block: false, alt: "1250")[$1250$]], )) + Evaluate the function at #math.equation(block: false, alt: "f open parenthesis 2 , 5 close parenthesis")[$f ( 2 , 5 )$]. What does your answer tell you about beams? + Is it true that #math.equation(block: false, alt: "f open parenthesis 2 , 5 close parenthesis equals f open parenthesis 5 , 2 close parenthesis")[$" " f ( 2 , 5 ) = f ( 5 , 2 )$]? + Consider the row corresponding to a width of #math.equation(block: false, alt: "3")[$3$] cm. How does the load depend on the depth? #solutionbox[ + In the row for #math.equation(block: false, alt: "2")[$2$] cm and the column for #math.equation(block: false, alt: "5")[$5$] cm, we find that #math.equation(block: false, alt: "f open parenthesis 2 , 5 close parenthesis equals 500")[$f ( 2 , 5 ) = 500$]. A beam of width #math.equation(block: false, alt: "2")[$2$] cm and depth #math.equation(block: false, alt: "5")[$5$] cm can support a maximum load of #math.equation(block: false, alt: "500")[$500$] kilograms. + In the row for #math.equation(block: false, alt: "5")[$5$] cm and the column for #math.equation(block: false, alt: "2")[$2$] cm, we find that #math.equation(block: false, alt: "f open parenthesis 5 , 2 close parenthesis equals 200")[$f ( 5 , 2 ) = 200$], so #math.equation(block: false, alt: "f open parenthesis 2 , 5 close parenthesis not equal to f open parenthesis 5 , 2 close parenthesis")[$f ( 2 , 5 ) ≠ f ( 5 , 2 )$]. + Using the row for width #math.equation(block: false, alt: "3")[$3$] cm, we make a new table showing the relationship between load and depth. The increase in load for each increase of #math.equation(block: false, alt: "1")[$1$] cm in depth is not a constant, so the graph shown below is not a straight line. #linebreak() The curve does pass through the origin, so perhaps the data describe direct variation with a power of depth. If we try the equation #math.equation(block: false, alt: "L equals k d squared")[$L = k d^(2)$] and use the point #math.equation(block: false, alt: "open parenthesis 1 , 30 close parenthesis")[$( 1 , 30 )$], we find that #math.equation(block: false, alt: "30 equals k times 1 squared")[$" " 30 = k ⋅ 1^(2)$], so #math.equation(block: false, alt: "k equals 30")[$k = 30$]. You can check that the equation #math.equation(block: false, alt: "L equals 30 d squared")[$L = 30 d^(2)$] does fit the rest of the data points. #figure(table( columns: 2, align: left, inset: 6pt, table.header([Depth], [Load]), [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "120")[$120$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "270")[$270$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "480")[$480$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "750")[$750$]], )) #figure(figph[load vs depth], alt: "load vs depth", caption: none) ] ] #figure(table( columns: 7, align: left, inset: 6pt, [Maximum Load (kilograms)], [], [], [], [], [], [], [], [Depth (cm)], [], [], [], [], [], [Width (cm)], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "90")[$90$]], [#math.equation(block: false, alt: "160")[$160$]], [#math.equation(block: false, alt: "250")[$250$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "80")[$80$]], [#math.equation(block: false, alt: "180")[$180$]], [#math.equation(block: false, alt: "320")[$320$]], [#math.equation(block: false, alt: "500")[$500$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "120")[$120$]], [#math.equation(block: false, alt: "270")[$270$]], [#math.equation(block: false, alt: "480")[$480$]], [#math.equation(block: false, alt: "750")[$750$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "160")[$160$]], [#math.equation(block: false, alt: "360")[$360$]], [#math.equation(block: false, alt: "640")[$640$]], [#math.equation(block: false, alt: "1000")[$1000$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "200")[$200$]], [#math.equation(block: false, alt: "450")[$450$]], [#math.equation(block: false, alt: "800")[$800$]], [#math.equation(block: false, alt: "1250")[$1250$]], )) + For the table in the previous example, consider the column corresponding to a beam depth of #math.equation(block: false, alt: "3")[$3$] cm. Graph #math.equation(block: false, alt: "L")[$L$] as a function of #math.equation(block: false, alt: "w")[$w$] when the depth is constant at #math.equation(block: false, alt: "d equals 3")[$d = 3$]. + Find a formula for #math.equation(block: false, alt: "L")[$L$] as a function of #math.equation(block: false, alt: "w")[$w$] for #math.equation(block: false, alt: "d equals 3")[$d = 3$]. #linebreak() #math.equation(block: false, alt: "L equals")[$L =$]\_\_\_\_\_ #solutionbox[ + A graph of the line is below. + #math.equation(block: false, alt: "L equals 90 w")[$L = 90 w$] ] #figure(figph[line], alt: "line", caption: none) + For the table in the previous example, consider the column corresponding to a beam depth of #math.equation(block: false, alt: "3")[$3$] cm. Graph #math.equation(block: false, alt: "L")[$L$] as a function of #math.equation(block: false, alt: "w")[$w$] when the depth is constant at #math.equation(block: false, alt: "d equals 3")[$d = 3$]. + Find a formula for #math.equation(block: false, alt: "L")[$L$] as a function of #math.equation(block: false, alt: "w")[$w$] for #math.equation(block: false, alt: "d equals 3")[$d = 3$]. #solutionbox[ + #figure(figph[line], alt: "line", caption: none) + #math.equation(block: false, alt: "L equals 90 w")[$L = 90 w$] ] In Practice 3, you should find that the load varies directly with width when the depth is #math.equation(block: false, alt: "3")[$3$] centimeters. In fact, the load varies directly with width for any fixed depth. In Practice 3, you should find that the load varies directly with width when the depth is #math.equation(block: false, alt: "3")[$3$] centimeters. In fact, the load varies directly with width for any fixed depth. In Example, we saw that the load varies with the square of depth when the width is #math.equation(block: false, alt: "3")[$3$] centimeters, and this relationship also holds for any value of #math.equation(block: false, alt: "w")[$w$]. Consequently, we can find a constant #math.equation(block: false, alt: "k")[$k$] such that #math.equation(block: true, alt: "load equals k times width times depth squared")[$"load" = k ⋅ "width" ⋅ "depth"^(2)$] This relationship between variables is an example of #strong[joint variation]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Joint Variation] - We say that #math.equation(block: false, alt: "z")[$z$] #strong[varies jointly] with #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] if #math.equation(block: true, alt: "z equals k x y , k not equal to 0")[$z = k x y , " " " " " " k ≠ 0$] - We say that #math.equation(block: false, alt: "z")[$z$] varies directly with #math.equation(block: false, alt: "x")[$x$] and inversely with #math.equation(block: false, alt: "y")[$y$] if #math.equation(block: true, alt: "z equals k the fraction x over y , k not equal to 0 , y not equal to 0")[$z = k display(frac(x, y)) , " " " " " " k ≠ 0 , " " " " y ≠ 0$] ] Which of the following functions represents joint variation, where #math.equation(block: false, alt: "k")[$k$] is a constant? \_\_\_\_\_ #solutionbox[ #math.equation(block: true, alt: "H equals k s t")[$H = k s t$] ] Which of the following functions represents joint variation, where #math.equation(block: false, alt: "k")[$k$] is a constant? + #math.equation(block: false, alt: "H equals k s t")[$H = k s t$] + #math.equation(block: false, alt: "L equals m plus k n")[$L = m + k n$] + #math.equation(block: false, alt: "y equals k r squared")[$y = k r^(2)$] + #math.equation(block: false, alt: "Z equals k open parenthesis x plus y close parenthesis")[$Z = k ( x + y )$] #examplebox("Example 4")[][ Find a formula for load as a function of width and depth for the data in Example. #solutionbox[ The function we want has the form #math.equation(block: true, alt: "L equals f open parenthesis w , d close parenthesis equals k w d squared")[$L = f ( w , d ) = k w d^(2)$] for some value of #math.equation(block: false, alt: "k")[$k$]. We use the fact that #math.equation(block: false, alt: "L equals 10")[$L = 10$] when #math.equation(block: false, alt: "w equals 1")[$w = 1$] and #math.equation(block: false, alt: "d equals 1")[$d = 1$]. Then #math.equation(block: true, alt: "10 equals k open parenthesis 1 close parenthesis open parenthesis 1 squared close parenthesis")[$10 = k ( 1 ) ( 1^(2) )$] so #math.equation(block: false, alt: "k equals 10")[$k = 10$]. The formula for load as a function of width and depth is #math.equation(block: true, alt: "L equals 10 w d squared")[$L = 10 w d^(2)$] You can check that this formula works for all the values in the table. ] ] The cost, #math.equation(block: false, alt: "C")[$C$], of tiling a rectangular floor depends on the dimensions (length and width) of the floor, so #math.equation(block: false, alt: "C equals f open parenthesis w , l close parenthesis")[$C = f ( w , l )$]. The table shows the costs in dollars for some dimensions. #figure(table( columns: 7, align: left, inset: 6pt, [Cost of Tiling a Floor], [], [], [], [], [], [], [], [Length (ft)], [], [], [], [], [], [Width (ft)], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "7")[$7$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "400")[$400$]], [#math.equation(block: false, alt: "480")[$480$]], [#math.equation(block: false, alt: "560")[$560$]], [#math.equation(block: false, alt: "640")[$640$]], [#math.equation(block: false, alt: "720")[$720$]], [#math.equation(block: false, alt: "800")[$800$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "480")[$480$]], [#math.equation(block: false, alt: "576")[$576$]], [#math.equation(block: false, alt: "672")[$672$]], [#math.equation(block: false, alt: "768")[$768$]], [#math.equation(block: false, alt: "864")[$864$]], [#math.equation(block: false, alt: "960")[$960$]], [#math.equation(block: false, alt: "7")[$7$]], [#math.equation(block: false, alt: "560")[$560$]], [#math.equation(block: false, alt: "672")[$672$]], [#math.equation(block: false, alt: "784")[$784$]], [#math.equation(block: false, alt: "896")[$896$]], [#math.equation(block: false, alt: "1008")[$1008$]], [#math.equation(block: false, alt: "1120")[$1120$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "640")[$640$]], [#math.equation(block: false, alt: "768")[$768$]], [#math.equation(block: false, alt: "896")[$896$]], [#math.equation(block: false, alt: "1024")[$1024$]], [#math.equation(block: false, alt: "1152")[$1152$]], [#math.equation(block: false, alt: "1280")[$1280$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "720")[$720$]], [#math.equation(block: false, alt: "864")[$864$]], [#math.equation(block: false, alt: "1008")[$1008$]], [#math.equation(block: false, alt: "1152")[$1152$]], [#math.equation(block: false, alt: "1296")[$1296$]], [#math.equation(block: false, alt: "1440")[$1440$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "800")[$800$]], [#math.equation(block: false, alt: "960")[$960$]], [#math.equation(block: false, alt: "1120")[$1120$]], [#math.equation(block: false, alt: "1280")[$1280$]], [#math.equation(block: false, alt: "1440")[$1440$]], [#math.equation(block: false, alt: "1600")[$1600$]], )) + Consider the row corresponding to #math.equation(block: false, alt: "6")[$6$] feet in width. Does cost vary directly with length? \_\_\_\_\_ + Consider the column corresponding to a length of #math.equation(block: false, alt: "10")[$10$] feet. Does the cost vary directly with width? \_\_\_\_\_ + Given that the cost varies jointly with the length and width of the floor, find a formula for #math.equation(block: false, alt: "C equals f open parenthesis w , l close parenthesis")[$C = f ( w , l )$]. #linebreak() #math.equation(block: false, alt: "C equals")[$C =$]\_\_\_\_\_ #solutionbox[ + Yes + Yes + #math.equation(block: false, alt: "C equals 16 w l")[$C = 16 w l$] ] The cost, #math.equation(block: false, alt: "C")[$C$], of tiling a rectangular floor depends on the dimensions (length and width) of the floor, so #math.equation(block: false, alt: "C equals f open parenthesis w , l close parenthesis")[$" " C = f ( w , l )$]. The table shows the costs in dollars for some dimensions. #figure(table( columns: 7, align: left, inset: 6pt, [Cost of Tiling a Floor], [], [], [], [], [], [], [], [Length (ft)], [], [], [], [], [], [Width (ft)], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "7")[$7$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "400")[$400$]], [#math.equation(block: false, alt: "480")[$480$]], [#math.equation(block: false, alt: "560")[$560$]], [#math.equation(block: false, alt: "640")[$640$]], [#math.equation(block: false, alt: "720")[$720$]], [#math.equation(block: false, alt: "800")[$800$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "480")[$480$]], [#math.equation(block: false, alt: "576")[$576$]], [#math.equation(block: false, alt: "672")[$672$]], [#math.equation(block: false, alt: "768")[$768$]], [#math.equation(block: false, alt: "864")[$864$]], [#math.equation(block: false, alt: "960")[$960$]], [#math.equation(block: false, alt: "7")[$7$]], [#math.equation(block: false, alt: "560")[$560$]], [#math.equation(block: false, alt: "672")[$672$]], [#math.equation(block: false, alt: "784")[$784$]], [#math.equation(block: false, alt: "896")[$896$]], [#math.equation(block: false, alt: "1008")[$1008$]], [#math.equation(block: false, alt: "1120")[$1120$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "640")[$640$]], [#math.equation(block: false, alt: "768")[$768$]], [#math.equation(block: false, alt: "896")[$896$]], [#math.equation(block: false, alt: "1024")[$1024$]], [#math.equation(block: false, alt: "1152")[$1152$]], [#math.equation(block: false, alt: "1280")[$1280$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "720")[$720$]], [#math.equation(block: false, alt: "864")[$864$]], [#math.equation(block: false, alt: "1008")[$1008$]], [#math.equation(block: false, alt: "1152")[$1152$]], [#math.equation(block: false, alt: "1296")[$1296$]], [#math.equation(block: false, alt: "1440")[$1440$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "800")[$800$]], [#math.equation(block: false, alt: "960")[$960$]], [#math.equation(block: false, alt: "1120")[$1120$]], [#math.equation(block: false, alt: "1280")[$1280$]], [#math.equation(block: false, alt: "1440")[$1440$]], [#math.equation(block: false, alt: "1600")[$1600$]], )) + Consider the row corresponding to #math.equation(block: false, alt: "6")[$6$] feet in width. Does cost vary directly with length? + Consider the column corresponding to a length of #math.equation(block: false, alt: "10")[$10$] feet. Does the cost vary directly with width? + Given that the cost varies jointly with the length and width of the floor, find a formula for #math.equation(block: false, alt: "C equals f open parenthesis w , l close parenthesis")[$C = f ( w , l )$]. #solutionbox[ + Yes + Yes + #math.equation(block: false, alt: "C equals 16 w l")[$C = 16 w l$] ] Explain the difference between the symbols #math.equation(block: false, alt: "f open parenthesis a b close parenthesis")[$f ( a b )$] and #math.equation(block: false, alt: "f open parenthesis a , b close parenthesis")[$f ( a , b )$]. \_\_\_\_\_ Explain the difference between the symbols #math.equation(block: false, alt: "f open parenthesis a b close parenthesis")[$f ( a b )$] and #math.equation(block: false, alt: "f open parenthesis a , b close parenthesis")[$f ( a , b )$]. === Graphs It is possible to make graphs in three dimensions for functions of two variables, but we will not do that here. Instead, we will represent such functions graphically by holding one of the two variables constant. #examplebox("Example 5")[][ In Example, we found a formula for the load a beam can support, #math.equation(block: true, alt: "L equals 10 w d squared")[$L = 10 w d^(2)$] + Graph #math.equation(block: false, alt: "L")[$L$] as a function of #math.equation(block: false, alt: "w")[$w$] for #math.equation(block: false, alt: "d equals 1 , 2 , 3 ,")[$d = 1 , 2 , 3 ,$] and #math.equation(block: false, alt: "4")[$4$]. + Graph #math.equation(block: false, alt: "L")[$L$] as a function of #math.equation(block: false, alt: "d")[$d$] for #math.equation(block: false, alt: "w equals 1 , 2 , 3 ,")[$w = 1 , 2 , 3 ,$] and #math.equation(block: false, alt: "4")[$4$]. #solutionbox[ + We make four graphs on the same grid, one for each value of #math.equation(block: false, alt: "d")[$d$]: #math.equation(block: true, alt: "when d, equals 1 ,, L equals 10 w; when d, equals 2 ,, L equals 40 w; when d, equals 3 ,, L equals 90 w; when d, equals 4 ,, L equals 160 w")[$"when " d & = 1 , & & L = 10 w \ "when " d & = 2 , & & L = 40 w \ "when " d & = 3 , & & L = 90 w \ "when " d & = 4 , & & L = 160 w$] The graphs are shown at left below. We can see that #math.equation(block: false, alt: "L")[$L$] varies directly with the width of the beam for any fixed value of its depth.#figure(figph[graphs of load vs width], alt: "graphs of load vs width", caption: none) #figure(figph[graphs of load vs depth], alt: "graphs of load vs depth", caption: none) + We make one graph for each value of #math.equation(block: false, alt: "w")[$w$]: #math.equation(block: true, alt: "when w, equals 1 ,, L equals 10 d squared; when w, equals 2 ,, L equals 20 d squared; when w, equals 3 ,, L equals 30 d squared; when w, equals 4 ,, L equals 40 d squared")[$"when " w & = 1 , & & L = 10 d^(2) \ "when " w & = 2 , & & L = 20 d^(2) \ "when " w & = 3 , & & L = 30 d^(2) \ "when " w & = 4 , & & L = 40 d^(2)$] The graphs are shown at right above. For any fixed value of its width, #math.equation(block: false, alt: "L")[$L$] varies directly with the square of depth. ] ] We can represent a function of two variables by \_\_\_\_\_ #solutionbox[ all of the above. ] We can represent a function of two variables by + a table of values. + a graph in three dimensions. + several graphs in two dimensions. + all of the above. The period of a satellite orbiting the Earth varies directly with the radius of the orbit and inversely with the speed of the satellite. + Write a formula for the period, #math.equation(block: false, alt: "T")[$T$], as a function of orbital radius, #math.equation(block: false, alt: "r")[$r$], and velocity, #math.equation(block: false, alt: "v")[$v$]. Your formula should contain a constant of variation #math.equation(block: false, alt: "k")[$k$] (whose value we will determine later). #linebreak() #math.equation(block: false, alt: "T equals f open parenthesis r , v close parenthesis equals")[$T = f ( r , v ) =$]\_\_\_\_\_ + GPS satellites orbit at an altitude of #math.equation(block: false, alt: "20 , 200")[$20 , 200$] kilometers and a speed of 233 kilometers per minute. The period of a GPS satellite is 11 hours and 58 minutes. Find the constant of variation in your formula for #math.equation(block: false, alt: "T")[$T$]. (The radius of the Earth is 6360 km.) #linebreak() #math.equation(block: false, alt: "k approximately equals")[$k ≈$]\_\_\_\_\_ + Satellites in polar orbits are used to measure ozone concentrations in the atmosphere. One such satellite orbits at an altitude of 833 km and has a period of 101.2 minutes. What is the speed of this satellite? #linebreak() Speed: \_\_\_\_\_\_\_\_\_\_ + Graph #math.equation(block: false, alt: "T")[$T$] as a function of #math.equation(block: false, alt: "v")[$v$] for #math.equation(block: false, alt: "r equals 5000")[$r = 5000$], #math.equation(block: false, alt: "r equals 10 , 000")[$r = 10 , 000$], #math.equation(block: false, alt: "r equals 20 , 000")[$r = 20 , 000$], and #math.equation(block: false, alt: "r equals 30 , 000")[$r = 30 , 000$]. #solutionbox[ + #math.equation(block: false, alt: "T equals f open parenthesis r , v close parenthesis equals the fraction k r over v")[$T = f ( r , v ) = display(frac(k r, v))$] + We first convert the period to minutes, then we substitute values into our formula and solve for #math.equation(block: false, alt: "k")[$k$] to find that #math.equation(block: false, alt: "k approximately equals 6.3")[$k ≈ 6.3$]. (Actually, #math.equation(block: false, alt: "k equals 2 π")[$k = 2 π$].) + #math.equation(block: false, alt: "448")[$448$] km/min + A graph is below. ] #figure(figph[curves], alt: "curves", caption: none) The period of a satellite orbiting the Earth varies directly with the radius of the orbit and inversely with the speed of the satellite. + Write a formula for the period, #math.equation(block: false, alt: "T")[$T$], as a function of orbital radius, #math.equation(block: false, alt: "r")[$r$], and velocity, #math.equation(block: false, alt: "v")[$v$]. + GPS satellites orbit at an altitude of #math.equation(block: false, alt: "20 , 200")[$20 , 200$] kilometers and a speed of 232 kilometers per minute. The period of a GPS satellite is 11 hours and 58 minutes. Find the constant of variation in your formula for #math.equation(block: false, alt: "T")[$T$]. (The radius of the Earth is 6360 km.) + Satellites in polar orbits are used to measure ozone concentrations in the atmosphere. One such satellite orbits at an altitude of 833 km and has a period of 101.2 minutes. What is the speed of this satellite? + Graph #math.equation(block: false, alt: "T")[$T$] as a function of #math.equation(block: false, alt: "v")[$v$] for #math.equation(block: false, alt: "r equals 5000")[$r = 5000$], #math.equation(block: false, alt: "r equals 10 , 000")[$r = 10 , 000$], #math.equation(block: false, alt: "r equals 20 , 000")[$r = 20 , 000$], and #math.equation(block: false, alt: "r equals 30 , 000")[$r = 30 , 000$]. #solutionbox[ + #math.equation(block: false, alt: "T equals f open parenthesis r , v close parenthesis equals the fraction k r over v")[$T = f ( r , v ) = display(frac(k r, v))$] + We first convert the period to minutes, then we substitute values into our formula and solve for #math.equation(block: false, alt: "k")[$k$] to find that #math.equation(block: false, alt: "k approximately equals 6.3")[$k ≈ 6.3$]. (Actually, #math.equation(block: false, alt: "k equals 2 π")[$k = 2 π$].) + #math.equation(block: false, alt: "448")[$448$] km/min + #figure(figph[curves], alt: "curves", caption: none) ] === Section Summary ==== Vocabulary Look up the definitions of new terms in the Glossary. - Function of two variables - Joint variation ==== CONCEPTS + The notation #math.equation(block: false, alt: "z equals f open parenthesis x , y close parenthesis")[$z = f ( x , y )$] indicates that #math.equation(block: false, alt: "z")[$z$] is a function of two variables, #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$]. + We can use a table with rows and columns to display the output values for a function of two variables. + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Joint Variation] - We say that #math.equation(block: false, alt: "z")[$z$] #strong[varies jointly] with #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] if #math.equation(block: true, alt: "z equals k x y , k not equal to 0")[$z = k x y , " " " " " " k ≠ 0$] - We say that #math.equation(block: false, alt: "z")[$z$] varies directly with #math.equation(block: false, alt: "x")[$x$] and inversely with #math.equation(block: false, alt: "y")[$y$] if #math.equation(block: true, alt: "z equals k the fraction x over y , k not equal to 0 , y not equal to 0")[$z = k display(frac(x, y)) , " " " " " " k ≠ 0 , " " " " y ≠ 0$] ] + We can represent a function of two variables graphically by showing a set of graphs for several fixed values of one of the variables. ==== STUDY QUESTIONS + Explain the difference between the symbols #math.equation(block: false, alt: "f open parenthesis a b close parenthesis")[$f ( a b )$] and #math.equation(block: false, alt: "f open parenthesis a , b close parenthesis")[$f ( a , b )$]. + Why is it true that #math.equation(block: false, alt: "f open parenthesis a b close parenthesis equals f open parenthesis b a close parenthesis")[$f ( a b ) = f ( b a )$], but not usually true that #math.equation(block: false, alt: "f open parenthesis a , b close parenthesis equals f open parenthesis b , a close parenthesis")[$f ( a , b ) = f ( b , a )$]? + If #math.equation(block: false, alt: "z")[$z$] varies jointly with #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$], and we hold one of the input variables constant, what will the graph look like? + What is wrong with the statement "#math.equation(block: false, alt: "z")[$z$] varies jointly with #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$], so #math.equation(block: false, alt: "z equals f open parenthesis x y close parenthesis")[$z = f ( x y )$]? ==== SKILLS Practice each skill in the Homework problems listed. + Evaluate the formula for a function of two or more variables, and interpret the result: \#1–6 + Evaluate a function of two variables from a table: \#7–10 + Write a formula for joint variation: \#11–18 + Graph a function of two variables by fixing values of one of the variables: \#11, 12, 15, 16, 19, 20 === Homework 3.5 Melody Airlines charges \$#math.equation(block: false, alt: "129")[$129$] for a coach ticket from San Francisco to Seattle, and \$#math.equation(block: false, alt: "240")[$240$] for a first-class ticket. + Write a function of two variables for the revenue, #math.equation(block: false, alt: "R")[$R$], that Melody Airlines will collect from the flight. + The airplane has #math.equation(block: false, alt: "12")[$12$] first-class seats and #math.equation(block: false, alt: "24")[$24$] coach seats. What is the maximum revenue the airline can collect? Write your answer with function notation. + #math.equation(block: false, alt: "R equals f open parenthesis x , y close parenthesis equals 129 x plus 240 y")[$R = f ( x , y ) = 129 x + 240 y$] + #math.equation(block: false, alt: "f open parenthesis 24 , 12 close parenthesis equals 5976")[$f ( 24 , 12 ) = 5976$] dollars is the maximum revenue. A manufacturing firm calculates its profit (or loss) by subtracting the cost of production from its revenue. The firm can produce 100 items per week, with fixed costs (overhead) of \$#math.equation(block: false, alt: "500")[$500$]. + Write a function for the firm's weekly profit, #math.equation(block: false, alt: "P")[$P$], if they charge a price of #math.equation(block: false, alt: "p")[$p$] dollars per item and it costs them #math.equation(block: false, alt: "c")[$c$] dollars to produce each item. + If each item costs \$#math.equation(block: false, alt: "80")[$80$] to produce, what price should the firm charge in order to make a profit? Write your answer with function notation. Archaeologists can calculate the size of a pot from just a fragment, or sherd, of the original. If #math.equation(block: false, alt: "L")[$L$] and #math.equation(block: false, alt: "h")[$h$] are the dimensions of an arc of a circle, as shown in the figure, then the radius of the entire circle is given by the function #math.equation(block: true, alt: "r equals f open parenthesis L , h close parenthesis equals the fraction L squared over 8 h plus the fraction h over 2")[$r = f ( L , h ) = frac(L^(2), 8 h) + frac(h, 2)$] #figure(figph[pot], alt: "pot", caption: none) + A pottery sherd has dimensions #math.equation(block: false, alt: "L equals 4")[$L = 4$] inches and #math.equation(block: false, alt: "h equals the fraction 3 over 2")[$h = display(frac(3, 2))$] inch. What was the radius of the whole pot? Write your answer with function notation. + Does #math.equation(block: false, alt: "r")[$r$] vary directly with #math.equation(block: false, alt: "L squared")[$L^(2)$]? Why or why not? + Show that the formula gives the correct value for #math.equation(block: false, alt: "r")[$r$] when the sherd is actually a semicircle. Hint: What are the values of #math.equation(block: false, alt: "L")[$L$] and #math.equation(block: false, alt: "h")[$h$] in this case? + #math.equation(block: false, alt: "f open parenthesis 4 , the fraction 3 over 2 close parenthesis equals the fraction 25 over 12 approximately equals 2.1")[$f ( 4 , display(frac(3, 2)) ) = display(frac(25, 12)) ≈ 2.1$] inches + No: We do not have #math.equation(block: false, alt: "r equals k L squared")[$r = k L^(2)$] for any constant #math.equation(block: false, alt: "k")[$k$]. + When #math.equation(block: false, alt: "L equals 2 r")[$L = 2 r$] and #math.equation(block: false, alt: "h equals r")[$h = r$], #math.equation(block: false, alt: "f open parenthesis L , h close parenthesis equals r")[$f ( L , h ) = r$]. The surface area of a cylinder is a function of its diameter and height, #math.equation(block: true, alt: "S equals f open parenthesis d , h close parenthesis equals π d h plus the fraction π over 2 d squared")[$S = f ( d , h ) = π d h + frac(π, 2) d^(2)$] + What is the surface area of a cylindrical oatmeal container with diameter #math.equation(block: false, alt: "4")[$4$] inches and height #math.equation(block: false, alt: "7")[$7$] inches? Write your answer with function notation. + Write a formula in terms of #math.equation(block: false, alt: "d")[$d$] for the surface area of a cylinder whose height is equal to its diameter. + How does the surface area of the cylinder in part (b) compare with the surface area of a sphere of the same diameter? Sketch both surfaces with the same center. The Dubois formula is used to estimate the surface area, #math.equation(block: false, alt: "S")[$S$], of a person in terms of his or her weight and height. A good estimate of surface area is critical to some forms of cancer treatment. In square centimeters, #math.equation(block: false, alt: "S")[$S$] is given by #math.equation(block: true, alt: "S equals f open parenthesis w , h close parenthesis equals 71.84 w to the power 0.425 h to the power 0.725")[$S = f ( w , h ) = 71.84 w^(0.425) h^(0.725)$] where #math.equation(block: false, alt: "w")[$w$] is in kilograms and #math.equation(block: false, alt: "h")[$h$] is in centimeters. + Use the Dubois formula to estimate the surface area of a person who weighs #math.equation(block: false, alt: "60")[$60$] kg and is #math.equation(block: false, alt: "160")[$160$] cm tall. + Does surface area increase more rapidly with weight or with height? + What percent increase in surface area does a #math.equation(block: false, alt: "10 %")[$10 upright(%)$] increase in weight produce? + #math.equation(block: false, alt: "16 , 220")[$16 , 220$] sq cm + Height + #math.equation(block: false, alt: "4.1 %")[$4.1 upright(%)$] In the 1970s, McNeill Alexander proposed a relationship between an animal's running speed, #math.equation(block: false, alt: "v")[$v$], its hip height, #math.equation(block: false, alt: "h")[$h$], and its stride length, #math.equation(block: false, alt: "s")[$s$]. If stride and hip height are measured in meters, the running speed is given in meters per second by #math.equation(block: true, alt: "v equals f open parenthesis h , s close parenthesis equals 0.78 s to the power 1.67 h to the power minus 1.17")[$v = f ( h , s ) = 0.78 s^(1.67) h^(− 1.17)$] + What is the speed of a racehorse whose hip height is #math.equation(block: false, alt: "1.6")[$1.6$] meters and stride length is #math.equation(block: false, alt: "7")[$7$] meters? + A cheetah can run at #math.equation(block: false, alt: "33")[$33$] meters per second. If its hip height is #math.equation(block: false, alt: "0.8")[$0.8$] meters, what is its stride length? + A pronghorn antelope has the same stride length as a cheetah and its hip height is #math.equation(block: false, alt: "12 %")[$12 upright(%)$] greater than the cheetah's. How does its running speed compare to a cheetah's? If you walk for exercise, the number of calories, #math.equation(block: false, alt: "C")[$C$], you burn per mile depends on your walking speed, #math.equation(block: false, alt: "s")[$s$], and your weight, #math.equation(block: false, alt: "w")[$w$]. + Write this fact in function notation. + Use the table to evaluate #math.equation(block: false, alt: "f open parenthesis 4.5 , 160 close parenthesis")[$f ( 4.5 , 160 )$] and explain its meaning. + Solve the inequality #math.equation(block: false, alt: "f open parenthesis s , 160 close parenthesis greater than 110")[$f ( s , 160 ) > 110$] and explain its meaning. + If you weigh #math.equation(block: false, alt: "140")[$140$] pounds, how fast should you walk to burn the most calories per mile? + How can you use the table to calculate how many calories you burn per hour while walking? #figure(table( columns: 8, align: left, inset: 6pt, [Calories Burned per Mile], [], [], [], [], [], [], [], [], [Weight (pounds)], [], [], [], [], [], [], [Speed #linebreak() (mph) #linebreak()], [#math.equation(block: false, alt: "100")[$100$]], [#math.equation(block: false, alt: "120")[$120$]], [#math.equation(block: false, alt: "140")[$140$]], [#math.equation(block: false, alt: "160")[$160$]], [#math.equation(block: false, alt: "180")[$180$]], [#math.equation(block: false, alt: "200")[$200$]], [#math.equation(block: false, alt: "220")[$220$]], [#math.equation(block: false, alt: "2.0")[$2.0$]], [#math.equation(block: false, alt: "65")[$65$]], [#math.equation(block: false, alt: "80")[$80$]], [#math.equation(block: false, alt: "93")[$93$]], [#math.equation(block: false, alt: "105")[$105$]], [#math.equation(block: false, alt: "120")[$120$]], [#math.equation(block: false, alt: "133")[$133$]], [#math.equation(block: false, alt: "145")[$145$]], [#math.equation(block: false, alt: "2.5")[$2.5$]], [#math.equation(block: false, alt: "62")[$62$]], [#math.equation(block: false, alt: "74")[$74$]], [#math.equation(block: false, alt: "88")[$88$]], [#math.equation(block: false, alt: "100")[$100$]], [#math.equation(block: false, alt: "112")[$112$]], [#math.equation(block: false, alt: "124")[$124$]], [#math.equation(block: false, alt: "138")[$138$]], [#math.equation(block: false, alt: "3.0")[$3.0$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "72")[$72$]], [#math.equation(block: false, alt: "83")[$83$]], [#math.equation(block: false, alt: "95")[$95$]], [#math.equation(block: false, alt: "108")[$108$]], [#math.equation(block: false, alt: "120")[$120$]], [#math.equation(block: false, alt: "132")[$132$]], [#math.equation(block: false, alt: "3.5")[$3.5$]], [#math.equation(block: false, alt: "59")[$59$]], [#math.equation(block: false, alt: "71")[$71$]], [#math.equation(block: false, alt: "83")[$83$]], [#math.equation(block: false, alt: "93")[$93$]], [#math.equation(block: false, alt: "107")[$107$]], [#math.equation(block: false, alt: "119")[$119$]], [#math.equation(block: false, alt: "130")[$130$]], [#math.equation(block: false, alt: "4.0")[$4.0$]], [#math.equation(block: false, alt: "59")[$59$]], [#math.equation(block: false, alt: "70")[$70$]], [#math.equation(block: false, alt: "81")[$81$]], [#math.equation(block: false, alt: "94")[$94$]], [#math.equation(block: false, alt: "105")[$105$]], [#math.equation(block: false, alt: "118")[$118$]], [#math.equation(block: false, alt: "129")[$129$]], [#math.equation(block: false, alt: "4.5")[$4.5$]], [#math.equation(block: false, alt: "69")[$69$]], [#math.equation(block: false, alt: "82")[$82$]], [#math.equation(block: false, alt: "97")[$97$]], [#math.equation(block: false, alt: "110")[$110$]], [#math.equation(block: false, alt: "122")[$122$]], [#math.equation(block: false, alt: "138")[$138$]], [#math.equation(block: false, alt: "151")[$151$]], [#math.equation(block: false, alt: "5.0")[$5.0$]], [#math.equation(block: false, alt: "77")[$77$]], [#math.equation(block: false, alt: "92")[$92$]], [#math.equation(block: false, alt: "108")[$108$]], [#math.equation(block: false, alt: "123")[$123$]], [#math.equation(block: false, alt: "138")[$138$]], [#math.equation(block: false, alt: "154")[$154$]], [#math.equation(block: false, alt: "169")[$169$]], [#math.equation(block: false, alt: "6.0")[$6.0$]], [#math.equation(block: false, alt: "86")[$86$]], [#math.equation(block: false, alt: "99")[$99$]], [#math.equation(block: false, alt: "114")[$114$]], [#math.equation(block: false, alt: "130")[$130$]], [#math.equation(block: false, alt: "147")[$147$]], [#math.equation(block: false, alt: "167")[$167$]], [#math.equation(block: false, alt: "190")[$190$]], [#math.equation(block: false, alt: "7.0")[$7.0$]], [#math.equation(block: false, alt: "96")[$96$]], [#math.equation(block: false, alt: "111")[$111$]], [#math.equation(block: false, alt: "128")[$128$]], [#math.equation(block: false, alt: "146")[$146$]], [#math.equation(block: false, alt: "165")[$165$]], [#math.equation(block: false, alt: "187")[$187$]], [#math.equation(block: false, alt: "212")[$212$]], )) + #math.equation(block: false, alt: "C equals f open parenthesis s , w close parenthesis")[$C = f ( s , w )$] + #math.equation(block: false, alt: "f open parenthesis 4.5 , 160 close parenthesis equals 110")[$f ( 4.5 , 160 ) = 110$], so someone walking #math.equation(block: false, alt: "4.5")[$4.5$] mph and weighing #math.equation(block: false, alt: "160")[$160$] pounds burns #math.equation(block: false, alt: "110")[$110$] calories per mile. + #math.equation(block: false, alt: "s greater than 4.5")[$s > 4.5$]. A person who weighs #math.equation(block: false, alt: "160")[$160$] pounds must walk faster than #math.equation(block: false, alt: "4.5")[$4.5$] mph in order to burn more than #math.equation(block: false, alt: "110")[$110$] calories per mile. + #math.equation(block: false, alt: "7")[$7$] mph + Find the row with your walking speed in the left column and move along that row until you are in the column with your weight at the top. The value in that row and column is the number of calories you burn per mile. The BMI (body mass index) is used to determine whether a person is a healthy weight, overweight, or obese. #math.equation(block: false, alt: "B")[$B$] is a function of height, #math.equation(block: false, alt: "h")[$h$], and weight, #math.equation(block: false, alt: "w")[$w$]. + Write this fact in function notation. + Use the table to evaluate #math.equation(block: false, alt: "f open parenthesis 68 , 150 close parenthesis ,")[$f ( 68 , 150 ) ,$] and explain its meaning. + A person is deemed overweight if his or her BMI is at least #math.equation(block: false, alt: "25")[$25$] but less than #math.equation(block: false, alt: "30")[$30$]. Write this fact in function notation. (A person is obese if the BMI is over #math.equation(block: false, alt: "30")[$30$].) + Solve the inequality #math.equation(block: false, alt: "f open parenthesis 66 , w close parenthesis less than 25")[$f ( 66 , w ) < 25$] and explain its meaning. + Solve the inequality #math.equation(block: false, alt: "f open parenthesis h , 200 close parenthesis greater than 30")[$f ( h , 200 ) > 30$] and explain its meaning. #figure(table( columns: 12, align: left, inset: 6pt, [Body Mass Index], [], [], [], [], [], [], [], [], [], [], [], [], [Weight (pounds)], [], [], [], [], [], [], [], [], [], [], [Height #linebreak() (inches) #linebreak()], [#math.equation(block: false, alt: "120")[$120$]], [#math.equation(block: false, alt: "130")[$130$]], [#math.equation(block: false, alt: "140")[$140$]], [#math.equation(block: false, alt: "150")[$150$]], [#math.equation(block: false, alt: "160")[$160$]], [#math.equation(block: false, alt: "170")[$170$]], [#math.equation(block: false, alt: "180")[$180$]], [#math.equation(block: false, alt: "190")[$190$]], [#math.equation(block: false, alt: "200")[$200$]], [#math.equation(block: false, alt: "210")[$210$]], [#math.equation(block: false, alt: "220")[$220$]], [#math.equation(block: false, alt: "58")[$58$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "29")[$29$]], [#math.equation(block: false, alt: "31")[$31$]], [#math.equation(block: false, alt: "33")[$33$]], [#math.equation(block: false, alt: "36")[$36$]], [#math.equation(block: false, alt: "38")[$38$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "42")[$42$]], [#math.equation(block: false, alt: "44")[$44$]], [#math.equation(block: false, alt: "46")[$46$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "23")[$23$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "29")[$29$]], [#math.equation(block: false, alt: "31")[$31$]], [#math.equation(block: false, alt: "33")[$33$]], [#math.equation(block: false, alt: "35")[$35$]], [#math.equation(block: false, alt: "37")[$37$]], [#math.equation(block: false, alt: "39")[$39$]], [#math.equation(block: false, alt: "41")[$41$]], [#math.equation(block: false, alt: "43")[$43$]], [#math.equation(block: false, alt: "62")[$62$]], [#math.equation(block: false, alt: "22")[$22$]], [#math.equation(block: false, alt: "24")[$24$]], [#math.equation(block: false, alt: "26")[$26$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "29")[$29$]], [#math.equation(block: false, alt: "31")[$31$]], [#math.equation(block: false, alt: "33")[$33$]], [#math.equation(block: false, alt: "35")[$35$]], [#math.equation(block: false, alt: "37")[$37$]], [#math.equation(block: false, alt: "38")[$38$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "64")[$64$]], [#math.equation(block: false, alt: "21")[$21$]], [#math.equation(block: false, alt: "22")[$22$]], [#math.equation(block: false, alt: "24")[$24$]], [#math.equation(block: false, alt: "26")[$26$]], [#math.equation(block: false, alt: "28")[$28$]], [#math.equation(block: false, alt: "29")[$29$]], [#math.equation(block: false, alt: "31")[$31$]], [#math.equation(block: false, alt: "33")[$33$]], [#math.equation(block: false, alt: "34")[$34$]], [#math.equation(block: false, alt: "36")[$36$]], [#math.equation(block: false, alt: "38")[$38$]], [#math.equation(block: false, alt: "66")[$66$]], [#math.equation(block: false, alt: "19")[$19$]], [#math.equation(block: false, alt: "21")[$21$]], [#math.equation(block: false, alt: "23")[$23$]], [#math.equation(block: false, alt: "24")[$24$]], [#math.equation(block: false, alt: "26")[$26$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "29")[$29$]], [#math.equation(block: false, alt: "31")[$31$]], [#math.equation(block: false, alt: "32")[$32$]], [#math.equation(block: false, alt: "34")[$34$]], [#math.equation(block: false, alt: "36")[$36$]], [#math.equation(block: false, alt: "68")[$68$]], [#math.equation(block: false, alt: "18")[$18$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "21")[$21$]], [#math.equation(block: false, alt: "23")[$23$]], [#math.equation(block: false, alt: "24")[$24$]], [#math.equation(block: false, alt: "26")[$26$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "29")[$29$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "32")[$32$]], [#math.equation(block: false, alt: "34")[$34$]], [#math.equation(block: false, alt: "70")[$70$]], [#math.equation(block: false, alt: "17")[$17$]], [#math.equation(block: false, alt: "19")[$19$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "22")[$22$]], [#math.equation(block: false, alt: "23")[$23$]], [#math.equation(block: false, alt: "24")[$24$]], [#math.equation(block: false, alt: "26")[$26$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "29")[$29$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "32")[$32$]], [#math.equation(block: false, alt: "72")[$72$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "18")[$18$]], [#math.equation(block: false, alt: "19")[$19$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "22")[$22$]], [#math.equation(block: false, alt: "23")[$23$]], [#math.equation(block: false, alt: "24")[$24$]], [#math.equation(block: false, alt: "26")[$26$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "28")[$28$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "74")[$74$]], [#math.equation(block: false, alt: "15")[$15$]], [#math.equation(block: false, alt: "17")[$17$]], [#math.equation(block: false, alt: "18")[$18$]], [#math.equation(block: false, alt: "19")[$19$]], [#math.equation(block: false, alt: "21")[$21$]], [#math.equation(block: false, alt: "22")[$22$]], [#math.equation(block: false, alt: "23")[$23$]], [#math.equation(block: false, alt: "24")[$24$]], [#math.equation(block: false, alt: "26")[$26$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "28")[$28$]], [#math.equation(block: false, alt: "76")[$76$]], [#math.equation(block: false, alt: "15")[$15$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "17")[$17$]], [#math.equation(block: false, alt: "18")[$18$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "21")[$21$]], [#math.equation(block: false, alt: "22")[$22$]], [#math.equation(block: false, alt: "23")[$23$]], [#math.equation(block: false, alt: "24")[$24$]], [#math.equation(block: false, alt: "26")[$26$]], [#math.equation(block: false, alt: "27")[$27$]], )) An #emph[amortization table] shows the monthly payments for a loan or mortgage. The table below gives monthly payments for a loan of \$#math.equation(block: false, alt: "100 , 000")[$100 , 000$]. The monthly payment, #math.equation(block: false, alt: "P")[$P$], is a function of the annual interest rate, #math.equation(block: false, alt: "r")[$r$], and the length of the loan, #math.equation(block: false, alt: "t")[$t$], in years. #figure(table( columns: 7, align: left, inset: 6pt, [Monthly Payment], [], [], [], [], [], [], [], [Length of loan (years)], [], [], [], [], [], [Interest #linebreak() rate #linebreak()], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "15")[$15$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "0.05")[$0.05$]], [#math.equation(block: false, alt: "1879")[$1879$]], [#math.equation(block: false, alt: "1056")[$1056$]], [#math.equation(block: false, alt: "788")[$788$]], [#math.equation(block: false, alt: "657")[$657$]], [#math.equation(block: false, alt: "582")[$582$]], [#math.equation(block: false, alt: "535")[$535$]], [#math.equation(block: false, alt: "0.06")[$0.06$]], [#math.equation(block: false, alt: "1924")[$1924$]], [#math.equation(block: false, alt: "1105")[$1105$]], [#math.equation(block: false, alt: "840")[$840$]], [#math.equation(block: false, alt: "713")[$713$]], [#math.equation(block: false, alt: "641")[$641$]], [#math.equation(block: false, alt: "597")[$597$]], [#math.equation(block: false, alt: "0.07")[$0.07$]], [#math.equation(block: false, alt: "1969")[$1969$]], [#math.equation(block: false, alt: "1154")[$1154$]], [#math.equation(block: false, alt: "894")[$894$]], [#math.equation(block: false, alt: "771")[$771$]], [#math.equation(block: false, alt: "703")[$703$]], [#math.equation(block: false, alt: "661")[$661$]], [#math.equation(block: false, alt: "0.08")[$0.08$]], [#math.equation(block: false, alt: "2014")[$2014$]], [#math.equation(block: false, alt: "1205")[$1205$]], [#math.equation(block: false, alt: "949")[$949$]], [#math.equation(block: false, alt: "831")[$831$]], [#math.equation(block: false, alt: "767")[$767$]], [#math.equation(block: false, alt: "729")[$729$]], [#math.equation(block: false, alt: "0.09")[$0.09$]], [#math.equation(block: false, alt: "2060")[$2060$]], [#math.equation(block: false, alt: "1257")[$1257$]], [#math.equation(block: false, alt: "1007")[$1007$]], [#math.equation(block: false, alt: "893")[$893$]], [#math.equation(block: false, alt: "833")[$833$]], [#math.equation(block: false, alt: "799")[$799$]], [#math.equation(block: false, alt: "0.10")[$0.10$]], [#math.equation(block: false, alt: "2107")[$2107$]], [#math.equation(block: false, alt: "1311")[$1311$]], [#math.equation(block: false, alt: "1066")[$1066$]], [#math.equation(block: false, alt: "957")[$957$]], [#math.equation(block: false, alt: "901")[$901$]], [#math.equation(block: false, alt: "870")[$870$]], )) + You would like to borrow \$#math.equation(block: false, alt: "100 , 000")[$100 , 000$] for #math.equation(block: false, alt: "20")[$20$] years. What interest rate, to the nearest percent, can you accept if your monthly payments must be no more than \$#math.equation(block: false, alt: "800")[$800$]? What interest rate can you accept if the loan is for #math.equation(block: false, alt: "30")[$30$] years? If #math.equation(block: false, alt: "P equals f open parenthesis r , t close parenthesis")[$P = f ( r , t )$], use function notation to write both of these questions as inequalities. + Suppose you borrow \$#math.equation(block: false, alt: "100 , 000")[$100 , 000$] for #math.equation(block: false, alt: "10")[$10$] years at #math.equation(block: false, alt: "10 %")[$10 upright(%)$] interest. Which would cause a greater reduction in your monthly payment: Reducing the interest rate by #math.equation(block: false, alt: "5 %")[$5 upright(%)$] or increasing the length of the loan by #math.equation(block: false, alt: "5")[$5$] years? + At a fixed interest rate of #math.equation(block: false, alt: "8 %")[$8 upright(%)$], is the monthly payment a linear function of the length of the loan? + For a fixed loan period of #math.equation(block: false, alt: "25")[$25$] years, is the monthly payment a linear function of interest rate? + Does a #math.equation(block: false, alt: "1 %")[$1 upright(%)$] increase in the interest rate have a greater affect on the monthly payment for a #math.equation(block: false, alt: "15")[$15$]-year loan or a #math.equation(block: false, alt: "30")[$30$]-year loan? + When is #math.equation(block: false, alt: "f open parenthesis r , 20 close parenthesis less than or equal to 800")[$f ( r , 20 ) ≤ 800$]? #math.equation(block: false, alt: "r less than or equal to 7 %")[$" " r ≤ 7 upright(%)$]; When is #math.equation(block: false, alt: "f open parenthesis r , 30 close parenthesis less than or equal to 800")[$f ( r , 30 ) ≤ 800$]? #math.equation(block: false, alt: "r less than or equal to 9 %")[$" " r ≤ 9 upright(%)$] + Reducing interest rate by #math.equation(block: false, alt: "5 %")[$5 upright(%)$] + No + No + #math.equation(block: false, alt: "30")[$30$]-year loan Warmer air can hold more moisture than cooler air. Relative humidity is the amount of moisture in the air, as a fraction of the saturation level at the current temperature. A common measure of humidity is the dewpoint: the temperature at which the current humidity would saturate the air, so that dew forms. The table gives dewpoints, #math.equation(block: false, alt: "D")[$D$], in #math.equation(block: false, alt: "∘")[$∘$]F, as a function of temperature, #math.equation(block: false, alt: "T")[$T$], and relative humidity, #math.equation(block: false, alt: "H")[$H$], #math.equation(block: false, alt: "D equals f open parenthesis T , H close parenthesis")[$D = f ( T , H )$]. #figure(table( columns: 12, align: left, inset: 6pt, [Dewpoint], [], [], [], [], [], [], [], [], [], [], [], [], [Relative humidity (#math.equation(block: false, alt: "%")[$upright(%)$])], [], [], [], [], [], [], [], [], [], [], [Temperature (#math.equation(block: false, alt: "∘")[$∘$]F)], [#math.equation(block: false, alt: "1")[$1$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "70")[$70$]], [#math.equation(block: false, alt: "80")[$80$]], [#math.equation(block: false, alt: "900")[$900$]], [#math.equation(block: false, alt: "100")[$100$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "minus 60")[$− 60$]], [#math.equation(block: false, alt: "minus 20")[$− 20$]], [#math.equation(block: false, alt: "minus 6")[$− 6$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "14")[$14$]], [#math.equation(block: false, alt: "18")[$18$]], [#math.equation(block: false, alt: "21")[$21$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "minus 53")[$− 53$]], [#math.equation(block: false, alt: "minus 12")[$− 12$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "11")[$11$]], [#math.equation(block: false, alt: "18")[$18$]], [#math.equation(block: false, alt: "23")[$23$]], [#math.equation(block: false, alt: "28")[$28$]], [#math.equation(block: false, alt: "31")[$31$]], [#math.equation(block: false, alt: "34")[$34$]], [#math.equation(block: false, alt: "37")[$37$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "minus 47")[$− 47$]], [#math.equation(block: false, alt: "minus 4")[$− 4$]], [#math.equation(block: false, alt: "11")[$11$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "32")[$32$]], [#math.equation(block: false, alt: "37")[$37$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "44")[$44$]], [#math.equation(block: false, alt: "47")[$47$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "minus 41")[$− 41$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "19")[$19$]], [#math.equation(block: false, alt: "29")[$29$]], [#math.equation(block: false, alt: "36")[$36$]], [#math.equation(block: false, alt: "41")[$41$]], [#math.equation(block: false, alt: "46")[$46$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "54")[$54$]], [#math.equation(block: false, alt: "57")[$57$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "70")[$70$]], [#math.equation(block: false, alt: "minus 35")[$− 35$]], [#math.equation(block: false, alt: "11")[$11$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "37")[$37$]], [#math.equation(block: false, alt: "45")[$45$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "55")[$55$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "64")[$64$]], [#math.equation(block: false, alt: "67")[$67$]], [#math.equation(block: false, alt: "70")[$70$]], [#math.equation(block: false, alt: "80")[$80$]], [#math.equation(block: false, alt: "minus 29")[$− 29$]], [#math.equation(block: false, alt: "19")[$19$]], [#math.equation(block: false, alt: "35")[$35$]], [#math.equation(block: false, alt: "46")[$46$]], [#math.equation(block: false, alt: "54")[$54$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "65")[$65$]], [#math.equation(block: false, alt: "69")[$69$]], [#math.equation(block: false, alt: "73")[$73$]], [#math.equation(block: false, alt: "77")[$77$]], [#math.equation(block: false, alt: "80")[$80$]], [#math.equation(block: false, alt: "90")[$90$]], [#math.equation(block: false, alt: "minus 23")[$− 23$]], [#math.equation(block: false, alt: "26")[$26$]], [#math.equation(block: false, alt: "43")[$43$]], [#math.equation(block: false, alt: "54")[$54$]], [#math.equation(block: false, alt: "62")[$62$]], [#math.equation(block: false, alt: "69")[$69$]], [#math.equation(block: false, alt: "74")[$74$]], [#math.equation(block: false, alt: "79")[$79$]], [#math.equation(block: false, alt: "83")[$83$]], [#math.equation(block: false, alt: "87")[$87$]], [#math.equation(block: false, alt: "90")[$90$]], )) + Estimate the relative humidity if the temperature is #math.equation(block: false, alt: "70 degrees")[$70^(∘)$]F and the dewpoint is #math.equation(block: false, alt: "40 degrees")[$40^(∘)$]F. Write your answer in function notation. + Does the dewpoint rise or fall with temperature? (Hint: Consider any column in the table, and notice how dewpoint changes with increasing temperature.) + Does the dewpoint rise or fall with humidity? (Hint: Consider any row in the table, and notice how dewpoint changes with increasing humidity.) + Suppose that the temperature is #math.equation(block: false, alt: "70 degrees")[$70^(∘)$]F and the relative humidity is #math.equation(block: false, alt: "70 %")[$70 upright(%)$]. Which would cause a larger change in dewpoint: a rise in temperature to #math.equation(block: false, alt: "80 degrees")[$80^(∘)$]F or an increase in humidity to #math.equation(block: false, alt: "80 %")[$80 upright(%)$]? + Does dewpoint change more rapidly with temperature when the humidity is low or when the humidity is high? The table shows automobile fuel efficiency, #math.equation(block: false, alt: "E")[$E$], as a function of gasoline used, #math.equation(block: false, alt: "g")[$g$], and miles driven, #math.equation(block: false, alt: "m")[$m$]. Values of #math.equation(block: false, alt: "E")[$E$] are rounded to tenths where necessary. #figure(table( columns: 9, align: left, inset: 6pt, [Fuel Efficiency], [], [], [], [], [], [], [], [], [], [Distance (miles)], [], [], [], [], [], [], [], [Gas #linebreak() (gallons) #linebreak()], [#math.equation(block: false, alt: "100")[$100$]], [#math.equation(block: false, alt: "150")[$150$]], [#math.equation(block: false, alt: "200")[$200$]], [#math.equation(block: false, alt: "250")[$250$]], [#math.equation(block: false, alt: "300")[$300$]], [#math.equation(block: false, alt: "350")[$350$]], [#math.equation(block: false, alt: "400")[$400$]], [#math.equation(block: false, alt: "450")[$450$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "12.5")[$12.5$]], [#math.equation(block: false, alt: "18.75")[$18.75$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "31.25")[$31.25$]], [#math.equation(block: false, alt: "37.5")[$37.5$]], [#math.equation(block: false, alt: "43.75")[$43.75$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "56.25")[$56.25$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "15")[$15$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "35")[$35$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "45")[$45$]], [#math.equation(block: false, alt: "12")[$12$]], [#math.equation(block: false, alt: "8.3")[$8.3$]], [#math.equation(block: false, alt: "12.5")[$12.5$]], [#math.equation(block: false, alt: "16.7")[$16.7$]], [#math.equation(block: false, alt: "20.8")[$20.8$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "29.2")[$29.2$]], [#math.equation(block: false, alt: "33.3")[$33.3$]], [#math.equation(block: false, alt: "37.5")[$37.5$]], [#math.equation(block: false, alt: "14")[$14$]], [#math.equation(block: false, alt: "7.1")[$7.1$]], [#math.equation(block: false, alt: "10.7")[$10.7$]], [#math.equation(block: false, alt: "14.3")[$14.3$]], [#math.equation(block: false, alt: "17.9")[$17.9$]], [#math.equation(block: false, alt: "21.4")[$21.4$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "28.6")[$28.6$]], [#math.equation(block: false, alt: "32.1")[$32.1$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "6.3")[$6.3$]], [#math.equation(block: false, alt: "9.4")[$9.4$]], [#math.equation(block: false, alt: "12.5")[$12.5$]], [#math.equation(block: false, alt: "15.6")[$15.6$]], [#math.equation(block: false, alt: "18.8")[$18.8$]], [#math.equation(block: false, alt: "21.9")[$21.9$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "28.1")[$28.1$]], [#math.equation(block: false, alt: "18")[$18$]], [#math.equation(block: false, alt: "5.6")[$5.6$]], [#math.equation(block: false, alt: "8.3")[$8.3$]], [#math.equation(block: false, alt: "11.1")[$11.1$]], [#math.equation(block: false, alt: "13.9")[$13.9$]], [#math.equation(block: false, alt: "16.7")[$16.7$]], [#math.equation(block: false, alt: "19.4")[$19.4$]], [#math.equation(block: false, alt: "22.2")[$22.2$]], [#math.equation(block: false, alt: "25")[$25$]], )) + Choose one row of the table and decide if #math.equation(block: false, alt: "E")[$E$] varies directly with #math.equation(block: false, alt: "m")[$m$] or inversely with #math.equation(block: false, alt: "m")[$m$]. Explain your method. + Choose one column of the table and decide if #math.equation(block: false, alt: "E")[$E$] varies directly with #math.equation(block: false, alt: "g")[$g$] or inversely with #math.equation(block: false, alt: "g")[$g$]. Explain your method. + Find the constant of variation and write #math.equation(block: false, alt: "E")[$E$] as a function of #math.equation(block: false, alt: "g")[$g$] and #math.equation(block: false, alt: "m")[$m$]. What are the units of #math.equation(block: false, alt: "E")[$E$]? + Sketch a graph of #math.equation(block: false, alt: "E")[$E$] as a function of #math.equation(block: false, alt: "m")[$m$] for #math.equation(block: false, alt: "g equals 8")[$g = 8$], #math.equation(block: false, alt: "10")[$10$], #math.equation(block: false, alt: "12")[$12$], and #math.equation(block: false, alt: "14")[$14$]. + Sketch a graph of #math.equation(block: false, alt: "E")[$E$] as a function of #math.equation(block: false, alt: "g")[$g$] for #math.equation(block: false, alt: "m equals 100")[$m = 100$], #math.equation(block: false, alt: "200")[$200$], #math.equation(block: false, alt: "300")[$300$], and #math.equation(block: false, alt: "400")[$400$]. + Direct variation: In each row, #math.equation(block: false, alt: "E equals k m")[$E = k m$] for some constant #math.equation(block: false, alt: "k")[$k$] that depends on the row. + Inverse variation: In each column, #math.equation(block: false, alt: "E equals the fraction c over g")[$E = display(frac(c, g))$] for some constant #math.equation(block: false, alt: "c")[$c$] that depends on the column. + #math.equation(block: false, alt: "E equals the fraction m over g")[$E = display(frac(m, g))$] miles/gallon + #figure(figph[four direct variation], alt: "four direct variation", caption: none) + #figure(figph[four inverse variation], alt: "four inverse variation", caption: none) The table shows the productivity, #math.equation(block: false, alt: "P")[$P$], of a manufacturing plant as a function of the number of items produced, #math.equation(block: false, alt: "I")[$I$], and the hours of labor used, #math.equation(block: false, alt: "w")[$w$]. Values of #math.equation(block: false, alt: "P")[$P$] are rounded to tenths where necessary. #figure(table( columns: 9, align: left, inset: 6pt, [Productivity], [], [], [], [], [], [], [], [], [], [Labor (hours)], [], [], [], [], [], [], [], [Items #linebreak() produced #linebreak()], [#math.equation(block: false, alt: "100")[$100$]], [#math.equation(block: false, alt: "150")[$150$]], [#math.equation(block: false, alt: "200")[$200$]], [#math.equation(block: false, alt: "250")[$250$]], [#math.equation(block: false, alt: "300")[$300$]], [#math.equation(block: false, alt: "350")[$350$]], [#math.equation(block: false, alt: "400")[$400$]], [#math.equation(block: false, alt: "450")[$450$]], [#math.equation(block: false, alt: "500")[$500$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "3.3")[$3.3$]], [#math.equation(block: false, alt: "2.5")[$2.5$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "1.7")[$1.7$]], [#math.equation(block: false, alt: "1.4")[$1.4$]], [#math.equation(block: false, alt: "1.3")[$1.3$]], [#math.equation(block: false, alt: "1.1")[$1.1$]], [#math.equation(block: false, alt: "600")[$600$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "2.4")[$2.4$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "1.7")[$1.7$]], [#math.equation(block: false, alt: "1.5")[$1.5$]], [#math.equation(block: false, alt: "1.3")[$1.3$]], [#math.equation(block: false, alt: "700")[$700$]], [#math.equation(block: false, alt: "7")[$7$]], [#math.equation(block: false, alt: "4.7")[$4.7$]], [#math.equation(block: false, alt: "3.5")[$3.5$]], [#math.equation(block: false, alt: "2.8")[$2.8$]], [#math.equation(block: false, alt: "2.3")[$2.3$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "1.75")[$1.75$]], [#math.equation(block: false, alt: "1.6")[$1.6$]], [#math.equation(block: false, alt: "800")[$800$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "5.3")[$5.3$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "3.2")[$3.2$]], [#math.equation(block: false, alt: "2.7")[$2.7$]], [#math.equation(block: false, alt: "2.3")[$2.3$]], [#math.equation(block: false, alt: "2")[$2$]], [#math.equation(block: false, alt: "1.8")[$1.8$]], [#math.equation(block: false, alt: "900")[$900$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "4.5")[$4.5$]], [#math.equation(block: false, alt: "3.6")[$3.6$]], [#math.equation(block: false, alt: "3")[$3$]], [#math.equation(block: false, alt: "26")[$26$]], [#math.equation(block: false, alt: "2.3")[$2.3$]], [#math.equation(block: false, alt: "2.1")[$2.1$]], [#math.equation(block: false, alt: "1000")[$1000$]], [#math.equation(block: false, alt: "10")[$10$]], [#math.equation(block: false, alt: "6.7")[$6.7$]], [#math.equation(block: false, alt: "5")[$5$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "3.3")[$3.3$]], [#math.equation(block: false, alt: "2.9")[$2.9$]], [#math.equation(block: false, alt: "2.5")[$2.5$]], [#math.equation(block: false, alt: "2.2")[$2.2$]], )) + Choose one row of the table and decide if #math.equation(block: false, alt: "P")[$P$] varies directly with #math.equation(block: false, alt: "w")[$w$] or inversely with #math.equation(block: false, alt: "w")[$w$]. Explain your method. + Choose one column of the table and decide if #math.equation(block: false, alt: "P")[$P$] varies directly with #math.equation(block: false, alt: "I")[$I$] or inversely with #math.equation(block: false, alt: "I")[$I$]. Explain your method. + Find the constant of variation and write #math.equation(block: false, alt: "P")[$P$] as a function of #math.equation(block: false, alt: "I")[$I$] and #math.equation(block: false, alt: "w")[$w$]. What are the units of #math.equation(block: false, alt: "P")[$P$]? + Sketch a graph of #math.equation(block: false, alt: "P")[$P$] as a function of #math.equation(block: false, alt: "w")[$w$] for #math.equation(block: false, alt: "I equals 500")[$I = 500$], #math.equation(block: false, alt: "600")[$600$], #math.equation(block: false, alt: "800")[$800$], and #math.equation(block: false, alt: "1000")[$1000$]. + Sketch a graph of #math.equation(block: false, alt: "P")[$P$] as a function of #math.equation(block: false, alt: "I")[$I$] for #math.equation(block: false, alt: "w equals 100")[$w = 100$], #math.equation(block: false, alt: "200")[$200$], #math.equation(block: false, alt: "300")[$300$], and #math.equation(block: false, alt: "400")[$400$]. The water gushing out of a fire hose exerts a backward force that the firefighter must control. This force, called the nozzle reaction, #math.equation(block: false, alt: "R")[$R$], is a function of the diameter, #math.equation(block: false, alt: "d")[$d$], of the nozzle and the water pressure, #math.equation(block: false, alt: "P")[$P$], at the nozzle. #figure(table( columns: 7, align: left, inset: 6pt, [Nozzle Reaction (lb)], [], [], [], [], [], [], [], [Water pressure (psi)], [], [], [], [], [], [Nozzle #linebreak() diameter #linebreak() (in) #linebreak()], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "70")[$70$]], [#math.equation(block: false, alt: "80")[$80$]], [#math.equation(block: false, alt: "1.00")[$1.00$]], [#math.equation(block: false, alt: "47.1")[$47.1$]], [#math.equation(block: false, alt: "62.8")[$62.8$]], [#math.equation(block: false, alt: "78.5")[$78.5$]], [#math.equation(block: false, alt: "94.2")[$94.2$]], [#math.equation(block: false, alt: "109.9")[$109.9$]], [#math.equation(block: false, alt: "125.6")[$125.6$]], [#math.equation(block: false, alt: "1.25")[$1.25$]], [#math.equation(block: false, alt: "73.6")[$73.6$]], [#math.equation(block: false, alt: "98.1")[$98.1$]], [#math.equation(block: false, alt: "122.7")[$122.7$]], [#math.equation(block: false, alt: "147.2")[$147.2$]], [#math.equation(block: false, alt: "171.7")[$171.7$]], [#math.equation(block: false, alt: "196.3")[$196.3$]], [#math.equation(block: false, alt: "1.50")[$1.50$]], [#math.equation(block: false, alt: "106")[$106$]], [#math.equation(block: false, alt: "141.3")[$141.3$]], [#math.equation(block: false, alt: "176.6")[$176.6$]], [#math.equation(block: false, alt: "212")[$212$]], [#math.equation(block: false, alt: "247.3")[$247.3$]], [#math.equation(block: false, alt: "282.6")[$282.6$]], [#math.equation(block: false, alt: "1.75")[$1.75$]], [#math.equation(block: false, alt: "144.2")[$144.2$]], [#math.equation(block: false, alt: "192.3")[$192.3$]], [#math.equation(block: false, alt: "240.4")[$240.4$]], [#math.equation(block: false, alt: "288.5")[$288.5$]], [#math.equation(block: false, alt: "336.6")[$336.6$]], [#math.equation(block: false, alt: "384.7")[$384.7$]], [#math.equation(block: false, alt: "2.00")[$2.00$]], [#math.equation(block: false, alt: "188.4")[$188.4$]], [#math.equation(block: false, alt: "251.2")[$251.2$]], [#math.equation(block: false, alt: "314")[$314$]], [#math.equation(block: false, alt: "376.8")[$376.8$]], [#math.equation(block: false, alt: "439.6")[$439.6$]], [#math.equation(block: false, alt: "502.4")[$502.4$]], )) + Show that #math.equation(block: false, alt: "R")[$R$] varies directly with #math.equation(block: false, alt: "P")[$P$]. + Show that #math.equation(block: false, alt: "R")[$R$] varies directly with a power of #math.equation(block: false, alt: "d")[$d$]. What is the power? + Find the constant of variation and write a formula for #math.equation(block: false, alt: "R")[$R$] as a function of #math.equation(block: false, alt: "d")[$d$] and #math.equation(block: false, alt: "P")[$P$]. + A typical fire hose has nozzle diameter #math.equation(block: false, alt: "2 the fraction 1 over 2")[$2 frac(1, 2)$] inches and nozzle pressure #math.equation(block: false, alt: "60")[$60$] psi. Use your formula to calculate the nozzle reaction. + In each row, #math.equation(block: false, alt: "R equals k p")[$R = k p$] for some constant #math.equation(block: false, alt: "k")[$k$] that depends on the row. + In each column, #math.equation(block: false, alt: "R equals c d squared")[$R = c d^(2)$] for some constant #math.equation(block: false, alt: "c")[$c$] that depends on the column. + #math.equation(block: false, alt: "R equals 1.57 d squared p")[$R = 1.57 d^(2) p$] + #math.equation(block: false, alt: "588.75")[$588.75$] pounds: When we keep #math.equation(block: false, alt: "p")[$p$] constant and double #math.equation(block: false, alt: "d")[$d$], #math.equation(block: false, alt: "R")[$R$] is multiplied by a factor of #math.equation(block: false, alt: "4")[$4$]. So the value at #math.equation(block: false, alt: "d equals 2 the fraction 1 over 2")[$d = 2 frac(1, 2)$], #math.equation(block: false, alt: "p equals 60")[$p = 60$] should be #math.equation(block: false, alt: "4")[$4$] times the value at #math.equation(block: false, alt: "d equals 1 the fraction 1 over 4")[$d = 1 frac(1, 4)$], #math.equation(block: false, alt: "p equals 60")[$p = 60$]. The resistance, #math.equation(block: false, alt: "R")[$R$], of a wire depends on its length, #math.equation(block: false, alt: "L")[$L$], and diameter, #math.equation(block: false, alt: "d")[$d$]. The table shows the resistance of copper wires of gauges from #math.equation(block: false, alt: "10")[$10$] to #math.equation(block: false, alt: "20")[$20$]. The diameters of the wires are given in mils, where #math.equation(block: false, alt: "1 mil equals 0.001 inch")[$1 " mil" = 0.001 " inch"$]. #figure(table( columns: 7, align: left, inset: 6pt, [Resistance (ohms)], [], [], [], [], [], [], [], [Diameter (mils)], [], [], [], [], [], [Length #linebreak() (ft) #linebreak()], [#math.equation(block: false, alt: "102")[$102$]], [#math.equation(block: false, alt: "81")[$81$]], [#math.equation(block: false, alt: "64")[$64$]], [#math.equation(block: false, alt: "51")[$51$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "32")[$32$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "0.0500")[$0.0500$]], [#math.equation(block: false, alt: "0.0793")[$0.0793$]], [#math.equation(block: false, alt: "0.1270")[$0.1270$]], [#math.equation(block: false, alt: "0.1999")[$0.1999$]], [#math.equation(block: false, alt: "0.325")[$0.325$]], [#math.equation(block: false, alt: "0.5078")[$0.5078$]], [#math.equation(block: false, alt: "100")[$100$]], [#math.equation(block: false, alt: "0.1000")[$0.1000$]], [#math.equation(block: false, alt: "0.1585")[$0.1585$]], [#math.equation(block: false, alt: "0.2539")[$0.2539$]], [#math.equation(block: false, alt: "0.3999")[$0.3999$]], [#math.equation(block: false, alt: "0.65")[$0.65$]], [#math.equation(block: false, alt: "1.0156")[$1.0156$]], [#math.equation(block: false, alt: "150")[$150$]], [#math.equation(block: false, alt: "0.1499")[$0.1499$]], [#math.equation(block: false, alt: "0.2378")[$0.2378$]], [#math.equation(block: false, alt: "0.3809")[$0.3809$]], [#math.equation(block: false, alt: "0.5998")[$0.5998$]], [#math.equation(block: false, alt: "0.975")[$0.975$]], [#math.equation(block: false, alt: "1.5234")[$1.5234$]], [#math.equation(block: false, alt: "200")[$200$]], [#math.equation(block: false, alt: "0.1999")[$0.1999$]], [#math.equation(block: false, alt: "0.3170")[$0.3170$]], [#math.equation(block: false, alt: "0.5078")[$0.5078$]], [#math.equation(block: false, alt: "0.7997")[$0.7997$]], [#math.equation(block: false, alt: "1.3")[$1.3$]], [#math.equation(block: false, alt: "2.0313")[$2.0313$]], )) + Show that #math.equation(block: false, alt: "R")[$R$] varies directly with #math.equation(block: false, alt: "L")[$L$]. + Show that #math.equation(block: false, alt: "R")[$R$] varies inversely with a power of #math.equation(block: false, alt: "d")[$d$]. What is the power? + Find the constant of variation and write a formula for #math.equation(block: false, alt: "R")[$R$] as a function of #math.equation(block: false, alt: "L")[$L$] and #math.equation(block: false, alt: "d")[$d$]. + Household current uses 12-gauge wire, with a diameter of 0.081 inches. Use your formula to calculate the resistance of a 100-foot length of 12-gauge wire, and verify with the table. In 2005, #emph[Popular Mechanics] tested the acceleration from rest for eight sports cars. #figure(table( columns: 4, align: left, inset: 6pt, table.header([Car], [Time #linebreak() (sec) #linebreak()], [Distance #linebreak() (ft) #linebreak()], [Acceleration #linebreak() (ft/sec #math.equation(block: false, alt: "2")[$2$]) #linebreak()]), [Mercedes-Benz E55 AMG], [#math.equation(block: false, alt: "5.06")[$5.06$]], [#math.equation(block: false, alt: "222.64")[$222.64$]], [#math.equation(block: false, alt: "17.39")[$17.39$]], [Lamborghini Gallardo], [#math.equation(block: false, alt: "4.68")[$4.68$]], [#math.equation(block: false, alt: "205.92")[$205.92$]], [#math.equation(block: false, alt: "18.80")[$18.80$]], [Chevrolet Corvette Z06], [#math.equation(block: false, alt: "4.58")[$4.58$]], [#math.equation(block: false, alt: "201.52")[$201.52$]], [#math.equation(block: false, alt: "19.21")[$19.21$]], [Mercedes-Benz SL600], [#math.equation(block: false, alt: "4.62")[$4.62$]], [#math.equation(block: false, alt: "203.29")[$203.29$]], [], [Porsche 911 GT2], [#math.equation(block: false, alt: "4.46")[$4.46$]], [#math.equation(block: false, alt: "196.24")[$196.24$]], [], [Dodge Viper SRT-10], [#math.equation(block: false, alt: "4.30")[$4.30$]], [#math.equation(block: false, alt: "189.20")[$189.20$]], [], [Saleen S7 Competition], [#math.equation(block: false, alt: "3.73")[$3.73$]], [#math.equation(block: false, alt: "164.12")[$164.12$]], [], [Ford Gran Torino], [#math.equation(block: false, alt: "3.43")[$3.43$]], [#math.equation(block: false, alt: "150.92")[$150.92$]], [], )) + Acceleration, #math.equation(block: false, alt: "a")[$a$], varies directly with distance, #math.equation(block: false, alt: "d")[$d$], and inversely with a power of time, #math.equation(block: false, alt: "t")[$t$]. Use the information in the table to write #math.equation(block: false, alt: "a")[$a$] as a function of #math.equation(block: false, alt: "d")[$d$] and #math.equation(block: false, alt: "t")[$t$]. + Use your formula to compute the acceleration of the other five cars. + Use your formula to plot #math.equation(block: false, alt: "a")[$a$] against #math.equation(block: false, alt: "d")[$d$] for #math.equation(block: false, alt: "t equals 1")[$t = 1$], #math.equation(block: false, alt: "2")[$2$], #math.equation(block: false, alt: "3")[$3$], and #math.equation(block: false, alt: "4")[$4$] seconds. + Use your formula to plot #math.equation(block: false, alt: "a")[$a$] against #math.equation(block: false, alt: "t")[$t$] for #math.equation(block: false, alt: "d equals 100")[$d = 100$], #math.equation(block: false, alt: "200")[$200$], #math.equation(block: false, alt: "300")[$300$], and #math.equation(block: false, alt: "400")[$400$] feet. + #math.equation(block: false, alt: "a equals the fraction 2 d over t squared")[$a = display(frac(2 d, t^(2)))$] + Mercedes-Benz: #math.equation(block: false, alt: "19.05 ft / sec squared")[$19.05 " " "ft" / "sec"^(2)$], Porsche: #math.equation(block: false, alt: "19.73 ft / sec squared")[$19.73 " " "ft" / "sec"^(2)$], Dodge: #math.equation(block: false, alt: "20.47 ft / sec squared")[$20.47 " " "ft" / "sec"^(2)$], Saleen: #math.equation(block: false, alt: "23.59 ft / sec squared")[$23.59 " " "ft" / "sec"^(2)$], Ford: #math.equation(block: false, alt: "25.66 ft / sec squared")[$25.66 " " "ft" / "sec"^(2)$] + #figure(figph[four lines], alt: "four lines", caption: none) + #figure(figph[four inverse square], alt: "four inverse square", caption: none) The rate of water flow, #math.equation(block: false, alt: "F")[$F$], through a fire hose is a function of the diameter, #math.equation(block: false, alt: "d")[$d$], of the nozzle and the nozzle pressure, #math.equation(block: false, alt: "p")[$p$]. #figure(table( columns: 7, align: left, inset: 6pt, [Water flow (gal/min)], [], [], [], [], [], [], [], [Water pressure (psi)], [], [], [], [], [], [Nozzle #linebreak() diameter #linebreak() (in) #linebreak()], [#math.equation(block: false, alt: "30")[$30$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "70")[$70$]], [#math.equation(block: false, alt: "80")[$80$]], [#math.equation(block: false, alt: "1.00")[$1.00$]], [#math.equation(block: false, alt: "164")[$164$]], [#math.equation(block: false, alt: "190")[$190$]], [#math.equation(block: false, alt: "212")[$212$]], [#math.equation(block: false, alt: "232")[$232$]], [#math.equation(block: false, alt: "251")[$251$]], [#math.equation(block: false, alt: "268")[$268$]], [#math.equation(block: false, alt: "1.25")[$1.25$]], [#math.equation(block: false, alt: "257")[$257$]], [#math.equation(block: false, alt: "296")[$296$]], [#math.equation(block: false, alt: "331")[$331$]], [#math.equation(block: false, alt: "363")[$363$]], [#math.equation(block: false, alt: "392")[$392$]], [#math.equation(block: false, alt: "419")[$419$]], [#math.equation(block: false, alt: "1.50")[$1.50$]], [#math.equation(block: false, alt: "370")[$370$]], [#math.equation(block: false, alt: "427")[$427$]], [#math.equation(block: false, alt: "477")[$477$]], [#math.equation(block: false, alt: "523")[$523$]], [#math.equation(block: false, alt: "565")[$565$]], [#math.equation(block: false, alt: "604")[$604$]], [#math.equation(block: false, alt: "1.75")[$1.75$]], [#math.equation(block: false, alt: "503")[$503$]], [#math.equation(block: false, alt: "581")[$581$]], [#math.equation(block: false, alt: "650")[$650$]], [#math.equation(block: false, alt: "712")[$712$]], [#math.equation(block: false, alt: "769")[$769$]], [#math.equation(block: false, alt: "822")[$822$]], [#math.equation(block: false, alt: "2.00")[$2.00$]], [#math.equation(block: false, alt: "657")[$657$]], [#math.equation(block: false, alt: "759")[$759$]], [#math.equation(block: false, alt: "849")[$849$]], [#math.equation(block: false, alt: "930")[$930$]], [#math.equation(block: false, alt: "1004")[$1004$]], [#math.equation(block: false, alt: "1073")[$1073$]], )) + Plot #math.equation(block: false, alt: "F")[$F$] as a function of #math.equation(block: false, alt: "d")[$d$] for #math.equation(block: false, alt: "p equals 30")[$p = 30$], #math.equation(block: false, alt: "40")[$40$], #math.equation(block: false, alt: "50")[$50$], and #math.equation(block: false, alt: "60")[$60$]. Which basic function do your graphs resemble? + Plot #math.equation(block: false, alt: "F")[$F$] as a function of #math.equation(block: false, alt: "p")[$p$] for #math.equation(block: false, alt: "d equals 1")[$d = 1$], #math.equation(block: false, alt: "1.25")[$1.25$], #math.equation(block: false, alt: "1.5")[$1.5$], and #math.equation(block: false, alt: "1.75")[$1.75$]. Which basic function do your graphs resemble? + Write a formula for #math.equation(block: false, alt: "F")[$F$] as a function of #math.equation(block: false, alt: "d")[$d$] and #math.equation(block: false, alt: "p")[$p$], and find the constant of variation. Check your formula against the table. Railroad engineers use a transition curve between straight sections of track and bends, which are designed as arcs of circles. The length, #math.equation(block: false, alt: "L")[$L$], of the transition curve varies directly with the cube of the train's speed, #math.equation(block: false, alt: "v")[$v$], and inversely with the radius, #math.equation(block: false, alt: "R")[$R$], of the circular arc. + On a section of track where the speed limit is #math.equation(block: false, alt: "50")[$50$] mph, a circular bend has a radius of #math.equation(block: false, alt: "2000")[$2000$] feet, and the transition curve is #math.equation(block: false, alt: "200")[$200$] feet long. Find the constant of variation and write a formula for #math.equation(block: false, alt: "L")[$L$] as a function of #math.equation(block: false, alt: "v")[$v$] and #math.equation(block: false, alt: "R")[$R$]. + If the speed limit is increased by #math.equation(block: false, alt: "20 %")[$20 upright(%)$], how is the length of the transition curve affected? + If the radius of the bend is increased by #math.equation(block: false, alt: "20 %")[$20 upright(%)$], how is the length of the transition curve affected? + #math.equation(block: false, alt: "L equals the fraction 3.2 v cubed over R")[$L = display(frac(3.2 v^(3), R))$] + Increased by #math.equation(block: false, alt: "72.8 %")[$72.8 upright(%)$] + Decreased by #math.equation(block: false, alt: "16 the fraction 2 over 3 %")[$16 frac(2, 3) upright(%)$] The density, #math.equation(block: false, alt: "D")[$D$], of a planet varies directly with its mass, #math.equation(block: false, alt: "M")[$M$], and inversely with the cube of its radius, #math.equation(block: false, alt: "r")[$r$]. #figure(table( columns: 4, align: left, inset: 6pt, table.header([Planet], [Radius #linebreak() (km) #linebreak()], [Mass #linebreak() (#math.equation(block: false, alt: "10 to the power 20 kg")[$10^(20) " kg"$]) #linebreak()], [Density #linebreak() (#math.equation(block: false, alt: "kg / m cubed")[$"kg" / "m"^(3)$]) #linebreak()]), [Mercury], [#math.equation(block: false, alt: "2440")[$2440$]], [#math.equation(block: false, alt: "3302")[$3302$]], [], [Venus], [#math.equation(block: false, alt: "6052")[$6052$]], [#math.equation(block: false, alt: "48 , 690")[$48 , 690$]], [], [Earth], [#math.equation(block: false, alt: "6378")[$6378$]], [#math.equation(block: false, alt: "59 , 740")[$59 , 740$]], [#math.equation(block: false, alt: "5497")[$5497$]], [Mars], [#math.equation(block: false, alt: "3397")[$3397$]], [#math.equation(block: false, alt: "6419")[$6419$]], [], [Jupiter], [#math.equation(block: false, alt: "71 , 490")[$71 , 490$]], [#math.equation(block: false, alt: "18 , 990 , 000")[$18 , 990 , 000$]], [], [Saturn], [#math.equation(block: false, alt: "60 , 270")[$60 , 270$]], [#math.equation(block: false, alt: "5 , 685 , 000")[$5 , 685 , 000$]], [], [Uranus], [#math.equation(block: false, alt: "25 , 560")[$25 , 560$]], [#math.equation(block: false, alt: "866 , 200")[$866 , 200$]], [], [Neptune], [#math.equation(block: false, alt: "24 , 765")[$24 , 765$]], [#math.equation(block: false, alt: "1 , 028 , 000")[$1 , 028 , 000$]], [], [Pluto], [#math.equation(block: false, alt: "1150")[$1150$]], [#math.equation(block: false, alt: "150")[$150$]], [], )) + Use the data for Earth to find the constant of variation, then write a formula for #math.equation(block: false, alt: "D")[$D$] as a function of #math.equation(block: false, alt: "M")[$M$] and #math.equation(block: false, alt: "r")[$r$]. + Calculate the densities of the other planets. + The planets are composed of three broad categories of materials: rocky materials, icy materials (including water), and the materials that dominate the sun, namely hydrogen and helium. The density of rock varies from #math.equation(block: false, alt: "3000")[$3000$] to #math.equation(block: false, alt: "8000 kg / m cubed")[$8000 " kg" / "m"^(3)$]. Which of the planets could be composed mainly of rock? Ammonia has many uses in industry and agriculture, including the production of fertilizers. It is produced in the laboratory from nitrogen and hydrogen, but the process requires high pressure and temperature for significant yield. The graph illustrates the relationship. (Source: Hunt and Sykes, 1984) #figure(figph[Yield vs pressure for five temperatures], alt: "Yield vs pressure for five temperatures", caption: none) + Complete the table showing the yield of ammonia, as a percent of the gas mixture leaving the reactor, at various pressures and temperatures.#figure(table( columns: 9, align: left, inset: 6pt, [Percent Ammonia], [], [], [], [], [], [], [], [], [], [Pressure (atmospheres)], [], [], [], [], [], [], [], [Temperature #linebreak() (#math.equation(block: false, alt: "∘")[$∘$]C) #linebreak()], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "100")[$100$]], [#math.equation(block: false, alt: "150")[$150$]], [#math.equation(block: false, alt: "200")[$200$]], [#math.equation(block: false, alt: "250")[$250$]], [#math.equation(block: false, alt: "300")[$300$]], [#math.equation(block: false, alt: "350")[$350$]], [#math.equation(block: false, alt: "400")[$400$]], [#math.equation(block: false, alt: "350")[$350$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [#math.equation(block: false, alt: "400")[$400$]], [], [], [], [], [], [], [], [], [#math.equation(block: false, alt: "450")[$450$]], [], [], [], [], [], [], [], [], [#math.equation(block: false, alt: "500")[$500$]], [], [], [], [], [], [], [], [], [#math.equation(block: false, alt: "550")[$550$]], [], [], [], [], [], [], [], [], )) + What happens to the yield of ammonia if the pressure is held constant but the temperature is increased beyond #math.equation(block: false, alt: "350 degrees")[$350^(∘)$]C? + Sketch a graph of the yield of ammonia as a function of temperature when the pressure is #math.equation(block: false, alt: "300")[$300$] atmospheres. + #figure(table( columns: 9, align: left, inset: 6pt, [Percent Ammonia], [], [], [], [], [], [], [], [], [], [Pressure (atmospheres)], [], [], [], [], [], [], [], [Temperature #linebreak() (#math.equation(block: false, alt: "∘")[$∘$]C) #linebreak()], [#math.equation(block: false, alt: "50")[$50$]], [#math.equation(block: false, alt: "100")[$100$]], [#math.equation(block: false, alt: "150")[$150$]], [#math.equation(block: false, alt: "200")[$200$]], [#math.equation(block: false, alt: "250")[$250$]], [#math.equation(block: false, alt: "300")[$300$]], [#math.equation(block: false, alt: "350")[$350$]], [#math.equation(block: false, alt: "400")[$400$]], [#math.equation(block: false, alt: "350")[$350$]], [#math.equation(block: false, alt: "25")[$25$]], [#math.equation(block: false, alt: "38")[$38$]], [#math.equation(block: false, alt: "46")[$46$]], [#math.equation(block: false, alt: "53")[$53$]], [#math.equation(block: false, alt: "58")[$58$]], [#math.equation(block: false, alt: "62")[$62$]], [#math.equation(block: false, alt: "66")[$66$]], [#math.equation(block: false, alt: "68")[$68$]], [#math.equation(block: false, alt: "400")[$400$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "26")[$26$]], [#math.equation(block: false, alt: "33")[$33$]], [#math.equation(block: false, alt: "38")[$38$]], [#math.equation(block: false, alt: "45")[$45$]], [#math.equation(block: false, alt: "48")[$48$]], [#math.equation(block: false, alt: "53")[$53$]], [#math.equation(block: false, alt: "56")[$56$]], [#math.equation(block: false, alt: "450")[$450$]], [#math.equation(block: false, alt: "9")[$9$]], [#math.equation(block: false, alt: "17")[$17$]], [#math.equation(block: false, alt: "23")[$23$]], [#math.equation(block: false, alt: "28")[$28$]], [#math.equation(block: false, alt: "32")[$32$]], [#math.equation(block: false, alt: "37")[$37$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "43")[$43$]], [#math.equation(block: false, alt: "500")[$500$]], [#math.equation(block: false, alt: "6")[$6$]], [#math.equation(block: false, alt: "11")[$11$]], [#math.equation(block: false, alt: "16")[$16$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "23")[$23$]], [#math.equation(block: false, alt: "27")[$27$]], [#math.equation(block: false, alt: "29")[$29$]], [#math.equation(block: false, alt: "32")[$32$]], [#math.equation(block: false, alt: "550")[$550$]], [#math.equation(block: false, alt: "4")[$4$]], [#math.equation(block: false, alt: "8")[$8$]], [#math.equation(block: false, alt: "11")[$11$]], [#math.equation(block: false, alt: "14")[$14$]], [#math.equation(block: false, alt: "17")[$17$]], [#math.equation(block: false, alt: "19")[$19$]], [#math.equation(block: false, alt: "22")[$22$]], [#math.equation(block: false, alt: "24")[$24$]], )) + The ammonia yield decreases. + #figure(figph[yield vs temperature], alt: "yield vs temperature", caption: none) The graph shows the heat index, which combines air temperature and relative humidity to determine an apparent temperature, or what the temperature actually feels like. (Source: Ahrens, 1998) #figure(figph[Air temperature vs relatve humidity for seven heat indices], alt: "Air temperature vs relatve humidity for seven heat indices", caption: none) + Complete the table showing the heat index for various combinations of air temperature and relative humidity.#figure(table( columns: 7, align: left, inset: 6pt, [Heat Index], [], [], [], [], [], [], [], [Relative humidity (#math.equation(block: false, alt: "%")[$upright(%)$])], [], [], [], [], [], [Air #linebreak() temperature #linebreak() (#math.equation(block: false, alt: "∘")[$∘$]F) #linebreak()], [#math.equation(block: false, alt: "0")[$0$]], [#math.equation(block: false, alt: "20")[$20$]], [#math.equation(block: false, alt: "40")[$40$]], [#math.equation(block: false, alt: "60")[$60$]], [#math.equation(block: false, alt: "80")[$80$]], [#math.equation(block: false, alt: "100")[$100$]], [#math.equation(block: false, alt: "80")[$80$]], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [#math.equation(block: false, alt: "90")[$90$]], [], [], [], [], [], [], [#math.equation(block: false, alt: "100")[$100$]], [], [], [], [], [], [], [#math.equation(block: false, alt: "110")[$110$]], [], [], [], [], [], [], [#math.equation(block: false, alt: "120")[$120$]], [], [], [], [], [], [], )) + Complete the table showing the relative humidity at which the heat index is equal to the actual air temperature.#figure(table( columns: 6, align: left, inset: 6pt, table.header([Air Temperature (#math.equation(block: false, alt: "∘")[$∘$]F)], [#math.equation(block: false, alt: "80")[$80$]], [#math.equation(block: false, alt: "90")[$90$]], [#math.equation(block: false, alt: "100")[$100$]], [#math.equation(block: false, alt: "110")[$110$]], [#math.equation(block: false, alt: "120")[$120$]]), [Relative humidity (#math.equation(block: false, alt: "%")[$upright(%)$])], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], [$#hide($0000$)$], )) + Sketch a graph of the heat index as a function of air temperature if the relative humidity is #math.equation(block: false, alt: "70 %")[$70 upright(%)$].