#set document(title: "8.9 Use Direct and Inverse 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")) == 8.9#h(0.6em)Use Direct and Inverse Variation #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. If you miss a problem, go back to the section listed and review the material. Find the multiplicative inverse of #math.equation(block: false, alt: "−8")[$−8$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "minus the fraction 1 over 8")[$− frac(1, 8)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve for #math.equation(block: false, alt: "n")[$n$]: #math.equation(block: false, alt: "45 equals 20 n")[$45 = 20 n$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "n equals 2.25")[$n = 2.25$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Evaluate #math.equation(block: false, alt: "5 x squared")[$5 x^(2)$] when #math.equation(block: false, alt: "x equals 10")[$x = 10$]. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "500")[$500$] ] ] When two quantities are related by a proportion, we say they are #emph[proportional] to each other. Another way to express this relation is to talk about the #emph[variation] of the two quantities. We will discuss direct variation and inverse variation in this section. === Solve Direct Variation Problems Lindsay gets paid \$15 per hour at her job. If we let #emph[s] be her salary and #emph[h] be the number of hours she has worked, we could model this situation with the equation #math.equation(block: true, alt: "s equals 15 h")[$s = 15 h$]Lindsay’s salary is the product of a constant, 15, and the number of hours she works. We say that Lindsay’s salary #emph[varies directly] with the number of hours she works. Two variables vary directly if one is the product of a constant and the other. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Direct Variation] For any two variables #emph[x] and #emph[y], #emph[y] varies directly with #emph[x] if #math.equation(block: true, alt: "y equals k x , where k not equal to 0")[$y = k x , #h(0.2em) "where" #h(0.2em) k ≠ 0$]The constant #emph[k] is called the constant of variation. ] In applications using direct variation, generally we will know values of one pair of the variables and will be asked to find the equation that relates #emph[x] and #emph[y]. Then we can use that equation to find values of #emph[y] for other values of #emph[x]. #examplebox("Example 1")[How to Solve Direct Variation Problems][ If y varies directly with x and #math.equation(block: false, alt: "y equals 20")[$y = 20$] when #math.equation(block: false, alt: "x equals 8")[$x = 8$], find the equation that relates #emph[x] and #emph[y]. #solutionbox[ #figure(figph[The above image has 3 columns. The table shows the steps to solve direct variation problems. Step one is to write the formula for the direct variation. The direct variation formula is y equals k x. Then we get y equals k times x.], alt: "The above image has 3 columns. The table shows the steps to solve direct variation problems. Step one is to write the formula for the direct variation. The direct variation formula is y equals k x. Then we get y equals k times x.", caption: none) #figure(figph[Step two is substitute the given values for the variables. We are given y equals 20 and x equals 8. Then we have 20 equals k times 8.], alt: "Step two is substitute the given values for the variables. We are given y equals 20 and x equals 8. Then we have 20 equals k times 8.", caption: none) #figure(figph[Step three is to solve for the constant variation. Divide both sides of the equation by 8, then multiply. We now get 20 divided by 8 equals k. K equals 2.5.], alt: "Step three is to solve for the constant variation. Divide both sides of the equation by 8, then multiply. We now get 20 divided by 8 equals k. K equals 2.5.", caption: none) #figure(figph[Step four is to write the equation that relates x and y. Rewrite the general equation with the value we found k to get y equals 2 and five-tenths times x.], alt: "Step four is to write the equation that relates x and y. Rewrite the general equation with the value we found k to get y equals 2 and five-tenths times x.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ If #math.equation(block: false, alt: "y")[$y$] varies directly as #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y equals 3 , when x equals 10 .")[$y = 3 , #h(0.2em) "when" #h(0.2em) x = 10 .$] find the equation that relates #emph[x] and #emph[y]. #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 3 over 10 x")[$y = frac(3, 10) x$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ If #math.equation(block: false, alt: "y")[$y$] varies directly as #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y equals 12 when x equals 4")[$y = 12 #h(0.2em) "when" #h(0.2em) x = 4$] find the equation that relates #emph[x] and #emph[y]. #solutionbox[ #math.equation(block: true, alt: "y equals 3 x")[$y = 3 x$] ] ] We’ll list the steps below. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Solve direct variation problems.] + Write the formula for direct variation. + Substitute the given values for the variables. + Solve for the constant of variation. + Write the equation that relates x and y. ] Now we’ll solve a few applications of direct variation. #examplebox("Example 2")[][ When Raoul runs on the treadmill at the gym, the number of calories, #emph[c], he burns varies directly with the number of minutes, #emph[m], he uses the treadmill. He burned 315 calories when he used the treadmill for 18 minutes. + ⓐ Write the equation that relates #emph[c] and #emph[m]. + ⓑ How many calories would he burn if he ran on the treadmill for 25 minutes? #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [The number of calories, #math.equation(block: false, alt: "c")[$c$], varies directly with #linebreak() the number of minutes, #math.equation(block: false, alt: "m")[$m$], on the treadmill, #linebreak() and #math.equation(block: false, alt: "c equals 315")[$c = 315$] when #math.equation(block: false, alt: "m equals 18")[$m = 18$].]), [Write the formula for direct variation.], [#figure(figph[The image displays the algebraic equation y = kx, which represents a direct proportionality relationship where 'y' is directly proportional to 'x' with 'k' as the constant of proportionality.], alt: "The image displays the algebraic equation y = kx, which represents a direct proportionality relationship where 'y' is directly proportional to 'x' with 'k' as the constant of proportionality.", caption: none)], [We will use #math.equation(block: false, alt: "c")[$c$] in place of #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "m")[$m$] in place of #math.equation(block: false, alt: "x")[$x$].], [#figure(figph[A mathematical equation is displayed on a white background, reading 'c = km'.], alt: "A mathematical equation is displayed on a white background, reading 'c = km'.", caption: none)], [Substitute the given values for the variables.], [#figure(figph[The mathematical equation '315 = k ⋅ 18' is displayed, showing the number 315 equal to the variable k multiplied by the number 18. The numbers 315 and 18 are in a reddish-brown color, while the equals sign, variable k, and multiplication dot are in black.], alt: "The mathematical equation '315 = k ⋅ 18' is displayed, showing the number 315 equal to the variable k multiplied by the number 18. The numbers 315 and 18 are in a reddish-brown color, while the equals sign, variable k, and multiplication dot are in black.", caption: none)], [Solve for the constant of variation.], [#figure(figph[A mathematical equation shows '315/18 = (k \* 18)/18', where the number 18 is highlighted in red in the numerator of the right-hand side of the equation.], alt: "A mathematical equation shows '315/18 = (k * 18)/18', where the number 18 is highlighted in red in the numerator of the right-hand side of the equation.", caption: none)], [], [#figure(figph[The image displays a simple mathematical equation, '17.5 = k', where 17.5 is equated to the variable k, indicating that the value of k is seventeen and a half.], alt: "The image displays a simple mathematical equation, '17.5 = k', where 17.5 is equated to the variable k, indicating that the value of k is seventeen and a half.", caption: none)], [Write the equation that relates #math.equation(block: false, alt: "c")[$c$] and #math.equation(block: false, alt: "m")[$m$].], [#figure(figph[A mathematical equation is displayed on a white background, reading 'c = km' in black text.], alt: "A mathematical equation is displayed on a white background, reading 'c = km' in black text.", caption: none)], [Substitute in the constant of variation.], [#figure(figph[The image displays the equation 'c = 17.5m' in a simple, clear font on a white background, likely representing a variable 'c' equaling 17.5 meters.], alt: "The image displays the equation 'c = 17.5m' in a simple, clear font on a white background, likely representing a variable 'c' equaling 17.5 meters.", caption: none)], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [Find #math.equation(block: false, alt: "c")[$c$] when #math.equation(block: false, alt: "m equals 25")[$m = 25$].]), [Write the equation that relates #math.equation(block: false, alt: "c")[$c$] and#math.equation(block: false, alt: "m")[$m$].], [#figure(figph[The image displays the text 'c = 17.5m' in a simple, clear font on a white background, likely indicating a measurement or variable assignment.], alt: "The image displays the text 'c = 17.5m' in a simple, clear font on a white background, likely indicating a measurement or variable assignment.", caption: none)], [Substitute the given value for #math.equation(block: false, alt: "m")[$m$].], [#figure(figph[A mathematical equation displays 'c = 17.5(25)', with the number '25' highlighted in red.], alt: "A mathematical equation displays 'c = 17.5(25)', with the number '25' highlighted in red.", caption: none)], [Simplify.], [#figure(figph[The image displays the equation c = 437.5, rendered in a digital, bold font against a white background.], alt: "The image displays the equation c = 437.5, rendered in a digital, bold font against a white background.", caption: none)], [], [Raoul would burn 437.5 calories if he used the #linebreak() treadmill for 25 minutes.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The number of calories, #emph[c], burned varies directly with the amount of time, #emph[t,] spent exercising. Arnold burned 312 calories in 65 minutes exercising. + ⓐ Write the equation that relates #emph[c] and #emph[t]. + ⓑ How many calories would he burn if he exercises for 90 minutes? #solutionbox[ ⓐ #math.equation(block: false, alt: "c equals 4.8 t")[$c = 4.8 t$] ⓑ 432 calories ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The distance a moving body travels, #emph[d], varies directly with time, #emph[t], it moves. A train travels 100 miles in 2 hours ⓐ Write the equation that relates #emph[d] and #emph[t]. ⓑ How many miles would it travel in 5 hours? #solutionbox[ ⓐ #math.equation(block: false, alt: "d equals 50 t")[$d = 50 t$] ⓑ 250 miles ] ] In the previous example, the variables #emph[c] and #emph[m] were named in the problem. Usually that is not the case. We will have to name the variables in the next example as part of the solution, just like we do in most applied problems. #examplebox("Example 3")[][ The number of gallons of gas Eunice’s car uses varies directly with the number of miles she drives. Last week she drove 469.8 miles and used 14.5 gallons of gas. + ⓐ Write the equation that relates the number of gallons of gas used to the number of miles driven. + ⓑ How many gallons of gas would Eunice’s car use if she drove 1000 miles? #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [The number of gallons of gas varies directly with the number of miles driven.]), [First we will name the variables.], [Let #math.equation(block: false, alt: "g equals")[$g =$] number of gallons of gas. #linebreak() #math.equation(block: false, alt: "m equals")[$#h(1.3em) m =$] number of miles driven], [Write the formula for direct variation.], [#figure(figph[The image displays the algebraic equation 'y = kx' in a simple, clear font against a white background.], alt: "The image displays the algebraic equation 'y = kx' in a simple, clear font against a white background.", caption: none)], [We will use #math.equation(block: false, alt: "g")[$g$] in place of #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "m")[$m$] in place of #math.equation(block: false, alt: "x")[$x$].], [#figure(figph[The image shows the mathematical equation g = km on a plain white background.], alt: "The image shows the mathematical equation g = km on a plain white background.", caption: none)], [Substitute the given values for the variables.], [#figure(figph[The image displays a mathematical expression 'g = 14.5 when m = 469.8' in a sans-serif font, where the value of 'g' is highlighted in light blue and 'm' is in red.], alt: "The image displays a mathematical expression 'g = 14.5 when m = 469.8' in a sans-serif font, where the value of 'g' is highlighted in light blue and 'm' is in red.", caption: none)], [], [#figure(figph[A mathematical equation is displayed with '14.5' in light blue, an equals sign in black, 'k' in black, and '(469.8)' in red.], alt: "A mathematical equation is displayed with '14.5' in light blue, an equals sign in black, 'k' in black, and '(469.8)' in red.", caption: none)], [Solve for the constant of variation.], [#figure(figph[A mathematical equation is displayed on a white background, showing 14.5 divided by 469.8 equals k multiplied by 469.8, all divided by 469.8.], alt: "A mathematical equation is displayed on a white background, showing 14.5 divided by 469.8 equals k multiplied by 469.8, all divided by 469.8.", caption: none)], [We will round to the nearest thousandth.], [#figure(figph[The equation 0.031 = k is displayed on a white background, presenting a simple mathematical statement.], alt: "The equation 0.031 = k is displayed on a white background, presenting a simple mathematical statement.", caption: none)], [Write the equation that relates #math.equation(block: false, alt: "g")[$g$] and #math.equation(block: false, alt: "m")[$m$].], [#figure(figph[The mathematical equation 'g = km' is displayed in black text on a plain white background.], alt: "The mathematical equation 'g = km' is displayed in black text on a plain white background.", caption: none)], [Substitute in the constant of variation.], [#figure(figph[A mathematical expression states that g = 0.031m, displayed in black text against a plain white background.], alt: "A mathematical expression states that g = 0.031m, displayed in black text against a plain white background.", caption: none)], )) #linebreak() ⓑ #linebreak() #math.equation(block: false, alt: "Find g when m equals 1000 .; Write the equation that relates g and m ., g equals 0.031 m; Substitute the given value for m ., g equals 0.031 open parenthesis 1000 close parenthesis; Simplify., g equals 31; Eunice’s car would use 31 gallons of gas if she drove it 1,000 miles.")[$& & & "Find" #h(0.2em) g #h(0.2em) "when" #h(0.2em) m = 1000 . \ "Write the equation that relates" #h(0.2em) g #h(0.2em) "and" #h(0.2em) m . & & & g = 0.031 m \ "Substitute the given value for" #h(0.2em) m . & & & g = 0.031 ( 1000 ) \ "Simplify." & & & g = 31 \ & & & "Eunice’s car would use 31 gallons of gas if she drove it 1,000 miles."$] Notice that in this example, the units on the constant of variation are gallons/mile. In everyday life, we usually talk about miles/gallon. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The distance that Brad travels varies directly with the time spent traveling. Brad travelled 660 miles in 12 hours, + ⓐ Write the equation that relates the number of miles travelled to the time. + ⓑ How many miles could Brad travel in 4 hours? #solutionbox[ ⓐ #math.equation(block: false, alt: "m equals 55 h")[$m = 55 h$] ⓑ 220 miles ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The weight of a liquid varies directly as its volume. A liquid that weighs 24 pounds has a volume of 4 gallons. + ⓐ Write the equation that relates the weight to the volume. + ⓑ If a liquid has volume 13 gallons, what is its weight? #solutionbox[ ⓐ #math.equation(block: false, alt: "w equals 6 v")[$w = 6 v$] ⓑ 78 pounds ] ] In some situations, one variable varies directly with the square of the other variable. When that happens, the equation of direct variation is #math.equation(block: false, alt: "y equals k x squared")[$y = k #h(0.2em) x^(2)$]. We solve these applications just as we did the previous ones, by substituting the given values into the equation to solve for #emph[k]. #examplebox("Example 4")[][ The maximum load a beam will support varies directly with the square of the diagonal of the beam’s cross-section. A beam with diagonal 4” will support a maximum load of 75 pounds. + ⓐ Write the equation that relates the maximum load to the cross-section. + ⓑ What is the maximum load that can be supported by a beam with diagonal 8”? #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [The maximum load varies directly with the square of the diagonal of the cross-section.]), [Name the variables.], [Let #math.equation(block: false, alt: "L equals")[$L =$] maximum load. #linebreak() #math.equation(block: false, alt: "c equals")[$#h(1.8em) c =$] the diagonal of the cross-section], [Write the formula for direct variation, where #math.equation(block: false, alt: "y")[$y$] varies directly with the square of #math.equation(block: false, alt: "x")[$x$].], [#figure(figph[The image displays the algebraic equation y = kx^2, representing a parabolic relationship where y is directly proportional to the square of x, with k as the constant of proportionality.], alt: "The image displays the algebraic equation y = kx^2, representing a parabolic relationship where y is directly proportional to the square of x, with k as the constant of proportionality.", caption: none)], [We will use #math.equation(block: false, alt: "L")[$L$] in place of #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "c")[$c$] in place of #math.equation(block: false, alt: "x")[$x$].], [#figure(figph[A mathematical equation is displayed on a white background, showing L = kc^2 in black text.], alt: "A mathematical equation is displayed on a white background, showing L = kc^2 in black text.", caption: none)], [Substitute the given values for the variables.], [#figure(figph[The image displays a mathematical expression stating 'L = 75 when c = 4' against a white background.], alt: "The image displays a mathematical expression stating 'L = 75 when c = 4' against a white background.", caption: none)], [], [#figure(figph[A mathematical equation is displayed on a white background: '75 = k ⋅ 4²'. The number 75 is in a light blue hue, and '4²' is in red, while the rest of the equation is in black.], alt: "A mathematical equation is displayed on a white background: '75 = k ⋅ 4²'. The number 75 is in a light blue hue, and '4²' is in red, while the rest of the equation is in black.", caption: none)], [Solve for the constant of variation.], [#figure(figph[A mathematical equation shows a fraction 75/16 on the left side, equated to a fraction (k \* 16)/16 on the right side, indicating an algebraic step to solve for k.], alt: "A mathematical equation shows a fraction 75/16 on the left side, equated to a fraction (k * 16)/16 on the right side, indicating an algebraic step to solve for k.", caption: none)], [], [#figure(figph[A mathematical equation shows '4.6875 = k' on a white background.], alt: "A mathematical equation shows '4.6875 = k' on a white background.", caption: none)], [Write the equation that relates #math.equation(block: false, alt: "L")[$L$] and #math.equation(block: false, alt: "c")[$c$].], [#figure(figph[A clearly displayed algebraic equation, L = kc², on a plain white background, which is a mathematical expression showing L is directly proportional to the square of c, with k as the constant of proportionality.], alt: "A clearly displayed algebraic equation, L = kc², on a plain white background, which is a mathematical expression showing L is directly proportional to the square of c, with k as the constant of proportionality.", caption: none)], [Substitute in the constant of variation.], [#figure(figph[A mathematical equation stating L = 4.6875c^2 is displayed on a white background.], alt: "A mathematical equation stating L = 4.6875c^2 is displayed on a white background.", caption: none)], )) #linebreak() ⓑ #linebreak() #math.equation(block: false, alt: "Find L when c equals 8 .; Write the equation that relates L and c ., L equals 4.6875 c squared; Substitute the given value for c ., L equals 4.6875 open parenthesis 8 close parenthesis squared; Simplify., L equals 300; A beam with diagonal 8” could support; a maximum load of 300 pounds.")[$& & & #h(8em) "Find" #h(0.2em) L #h(0.2em) "when" #h(0.2em) c = 8 . \ "Write the equation that relates" #h(0.2em) L #h(0.2em) "and" #h(0.2em) c . & & & #h(8em) L = 4.6875 c^(2) \ "Substitute the given value for" #h(0.2em) c . & & & #h(8em) L = 4.6875 attach(( 8 ), t: 2) \ "Simplify." & & & #h(8em) L = 300 \ & & & #h(8em) "A beam with diagonal 8” could support" \ "a maximum load of 300 pounds."$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The distance an object falls is directly proportional to the square of the time it falls. A ball falls 144 feet in 3 seconds. + ⓐ Write the equation that relates the distance to the time. + ⓑ How far will an object fall in 4 seconds? #solutionbox[ ⓐ #math.equation(block: false, alt: "d equals 16 t squared")[$d = 16 t^(2)$] ⓑ 256 feet ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The area of a circle varies directly as the square of the radius. A circular pizza with a radius of 6 inches has an area of 113.04 square inches. + ⓐ Write the equation that relates the area to the radius. + ⓑ What is the area of a pizza with a radius of 9 inches? #solutionbox[ ⓐ #math.equation(block: false, alt: "A equals 3.14 r squared")[$A = 3.14 r^(2)$] ⓑ 254.34 square inches ] ] === Solve Inverse Variation Problems Many applications involve two variable that #emph[vary inversely]. As one variable increases, the other decreases. The equation that relates them is #math.equation(block: false, alt: "y equals the fraction k over x")[$y = frac(k, x)$]. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Inverse Variation] For any two variables #emph[x] and #emph[y], #emph[y] varies inversely with #emph[x] if #math.equation(block: true, alt: "y equals the fraction k over x , where k not equal to 0")[$y = frac(k, x) , #h(0.2em) "where" #h(0.2em) k ≠ 0$]The constant #emph[k] is called the constant of variation. ] The word ‘inverse’ in inverse variation refers to the multiplicative inverse. The multiplicative inverse of #emph[x] is #math.equation(block: false, alt: "the fraction 1 over x")[$frac(1, x)$]. We solve inverse variation problems in the same way we solved direct variation problems. Only the general form of the equation has changed. We will copy the procedure box here and just change ‘direct’ to ‘inverse’. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Solve inverse variation problems.] + Write the formula for inverse variation. + Substitute the given values for the variables. + Solve for the constant of variation. + Write the equation that relates x and y. ] #examplebox("Example 5")[][ If #emph[y] varies inversely with #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y equals 20")[$y = 20$] when #math.equation(block: false, alt: "x equals 8")[$x = 8$], find the equation that relates #emph[x] and #emph[y]. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Write the formula for inverse variation.], [#figure(figph[A mathematical equation displays 'y = k/x', illustrating the concept of inverse variation where y is inversely proportional to x, with k representing the constant of proportionality.], alt: "A mathematical equation displays 'y = k/x', illustrating the concept of inverse variation where y is inversely proportional to x, with k representing the constant of proportionality.", caption: none)]), [Substitute the given values for the variables.], [#figure(figph[The image shows a mathematical expression 'y = 20 when x = 8' written in black text against a white background. The number '20' is highlighted in a light blue color, and the number '8' is highlighted in red.], alt: "The image shows a mathematical expression 'y = 20 when x = 8' written in black text against a white background. The number '20' is highlighted in a light blue color, and the number '8' is highlighted in red.", caption: none)], [], [#figure(figph[A mathematical equation is displayed on a white background, reading '20 = k/8' with '20' in blue and '8' in red. The expression represents a simple algebraic problem.], alt: "A mathematical equation is displayed on a white background, reading '20 = k/8' with '20' in blue and '8' in red. The expression represents a simple algebraic problem.", caption: none)], [Solve for the constant of variation.], [#figure(figph[A mathematical equation is displayed, showing 8 multiplied by 20 on the left side, equaling 8 multiplied by the fraction k over 8 on the right side: 8(20) = 8(k/8).], alt: "A mathematical equation is displayed, showing 8 multiplied by 20 on the left side, equaling 8 multiplied by the fraction k over 8 on the right side: 8(20) = 8(k/8).", caption: none)], [], [#figure(figph[The equation 160 = k is displayed on a white background, indicating that the value of k is 160.], alt: "The equation 160 = k is displayed on a white background, indicating that the value of k is 160.", caption: none)], [Write the equation that relates #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$].], [#figure(figph[A mathematical equation displays 'y = k/x', illustrating the concept of inverse variation where y is inversely proportional to x, with k representing the constant of proportionality.], alt: "A mathematical equation displays 'y = k/x', illustrating the concept of inverse variation where y is inversely proportional to x, with k representing the constant of proportionality.", caption: none)], [Substitute in the constant of variation.], [#figure(figph[A mathematical equation is displayed on a white background, reading 'y = 160 / x'. The equation represents an inverse relationship between y and x.], alt: "A mathematical equation is displayed on a white background, reading 'y = 160 / x'. The equation represents an inverse relationship between y and x.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ If #math.equation(block: false, alt: "p")[$p$] varies inversely with #math.equation(block: false, alt: "q")[$q$] and #math.equation(block: false, alt: "p equals 30")[$p = 30$] when #math.equation(block: false, alt: "q equals 12")[$q = 12$] find the equation that relates #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "q .")[$q .$] #solutionbox[ #math.equation(block: true, alt: "p equals the fraction 360 over q")[$p = frac(360, q)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ If #math.equation(block: false, alt: "y")[$y$] varies inversely with #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y equals 8")[$y = 8$] when #math.equation(block: false, alt: "x equals 2")[$x = 2$] find the equation that relates #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$]. #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 16 over x")[$y = frac(16, x)$] ] ] #examplebox("Example 6")[][ The fuel consumption (mpg) of a car varies inversely with its weight. A car that weighs 3100 pounds gets 26 mpg on the highway. + ⓐ Write the equation of variation. + ⓑ What would be the fuel consumption of a car that weighs 4030 pounds? #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [The fuel consumption varies inversely with the weight.]), [First we will name the variables.], [Let #math.equation(block: false, alt: "f equals")[$f =$] fuel consumption. #linebreak() #math.equation(block: false, alt: "w equals")[$#h(1.8em) w =$] weight], [Write the formula for inverse variation.], [#figure(figph[The image displays a mathematical equation for inverse variation, written as y = k/x, where y is inversely proportional to x, and k is the constant of proportionality.], alt: "The image displays a mathematical equation for inverse variation, written as y = k/x, where y is inversely proportional to x, and k is the constant of proportionality.", caption: none)], [We will use #math.equation(block: false, alt: "f")[$f$] in place of #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "w")[$w$] in place of #math.equation(block: false, alt: "x")[$x$].], [#figure(figph[A mathematical formula is shown on a white background, reading f = k/w. This equation represents a relationship where f is inversely proportional to w, with k as the constant of proportionality.], alt: "A mathematical formula is shown on a white background, reading f = k/w. This equation represents a relationship where f is inversely proportional to w, with k as the constant of proportionality.", caption: none)], [Substitute the given values for the variables.], [#figure(figph[An equation showing 'f = 26' in light blue and 'w = 3100' in red.], alt: "An equation showing 'f = 26' in light blue and 'w = 3100' in red.", caption: none)], [], [#figure(figph[A mathematical equation is displayed on a white background, reading '26 = k / 3100'. The number '26' is in light blue, and '3100' is in red, with 'k' and the equals and division signs in black.], alt: "A mathematical equation is displayed on a white background, reading '26 = k / 3100'. The number '26' is in light blue, and '3100' is in red, with 'k' and the equals and division signs in black.", caption: none)], [Solve for the constant of variation.], [#figure(figph[A mathematical equation is displayed, showing 3100 multiplied by 26 on the left side, which equals 3100 multiplied by the fraction k over 3100 on the right side.], alt: "A mathematical equation is displayed, showing 3100 multiplied by 26 on the left side, which equals 3100 multiplied by the fraction k over 3100 on the right side.", caption: none)], [], [#figure(figph[The image displays a mathematical equation in black text on a white background, stating '80,600 = k'.], alt: "The image displays a mathematical equation in black text on a white background, stating '80,600 = k'.", caption: none)], [Write the equation that relates #math.equation(block: false, alt: "f")[$f$] and #math.equation(block: false, alt: "w")[$w$].], [#figure(figph[The image displays the mathematical formula f = k/w, where 'f' is equal to 'k' divided by 'w', set against a plain white background.], alt: "The image displays the mathematical formula f = k/w, where 'f' is equal to 'k' divided by 'w', set against a plain white background.", caption: none)], [Substitute in the constant of variation.], [#figure(figph[A mathematical equation is displayed, showing 'f equals 80,600 over w' on a white background, representing a formula likely used in algebra or physics.], alt: "A mathematical equation is displayed, showing 'f equals 80,600 over w' on a white background, representing a formula likely used in algebra or physics.", caption: none)], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "Find f when w equals 4030 .")[$"Find" #h(0.2em) f #h(0.2em) "when" #h(0.2em) w = 4030 .$]]), [Write the equation that relates #emph[f] and #emph[w].], [#math.equation(block: false, alt: "f equals the fraction 80,600 over w")[$f = frac(80","600, w)$]], [Substitute the given value for #emph[w].], [#math.equation(block: false, alt: "f equals the fraction 80,600 over 4030")[$f = frac(80","600, 4030)$]], [Simplify.], [#math.equation(block: false, alt: "f equals 20")[$f = 20$] #linebreak() A car that weighs 4030 pounds would #linebreak() have fuel consumption of 20 mpg.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ A car’s value varies inversely with its age. Elena bought a two-year-old car for \$20,000. ⓐ Write the equation of variation. ⓑ What will be the value of Elena’s car when it is 5 years old? #solutionbox[ ⓐ #math.equation(block: false, alt: "v equals the fraction 40,000 over a")[$v = frac(40","000, a)$] ⓑ \$8,000 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The time required to empty a pool varies inversely as the rate of pumping. It took Lucy 2.5 hours to empty her pool using a pump that was rated at 400 gpm (gallons per minute). + ⓐ Write the equation of variation. + ⓑ How long will it take her to empty the pool using a pump rated at 500 gpm? #solutionbox[ ⓐ #math.equation(block: false, alt: "t equals the fraction 1000 over r")[$t = frac(1000, r)$] ⓑ 2 hours ] ] #examplebox("Example 7")[][ The frequency of a guitar string varies inversely with its length. A 26” long string has a frequency of 440 vibrations per second. + ⓐ Write the equation of variation. + ⓑ How many vibrations per second will there be if the string’s length is reduced to 20” by putting a finger on a fret? #solutionbox[ ⓐ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [The frequency varies inversely with the length.]), [Name the variables.], [Let #math.equation(block: false, alt: "f equals")[$f =$] frequency. #linebreak() #math.equation(block: false, alt: "L equals")[$#h(1.5em) L =$] length], [Write the formula for inverse variation.], [#figure(figph[The image displays a mathematical equation for inverse proportionality, y = k/x, where 'y' is inversely proportional to 'x' with 'k' as the constant of proportionality.], alt: "The image displays a mathematical equation for inverse proportionality, y = k/x, where 'y' is inversely proportional to 'x' with 'k' as the constant of proportionality.", caption: none)], [We will use #math.equation(block: false, alt: "f")[$f$] in place of #math.equation(block: false, alt: "y")[$y$] and #math.equation(block: false, alt: "L")[$L$] in place of #math.equation(block: false, alt: "x")[$x$].], [#figure(figph[The mathematical formula shown is f = k/L, where 'f' is equal to 'k' divided by 'L'. This equation represents a relationship between three variables, often seen in physics or engineering contexts.], alt: "The mathematical formula shown is f = k/L, where 'f' is equal to 'k' divided by 'L'. This equation represents a relationship between three variables, often seen in physics or engineering contexts.", caption: none)], [Substitute the given values for the variables.], [#figure(figph[The equation f = 440 when L = 26 is displayed on a white background, with '440' in blue and '26' in red for emphasis.], alt: "The equation f = 440 when L = 26 is displayed on a white background, with '440' in blue and '26' in red for emphasis.", caption: none)], [], [#figure(figph[An algebraic equation is shown on a white background. The equation reads '440 = k/26'. The number 440 is in light blue, and the number 26 is in red, while 'k' and '=' are black.], alt: "An algebraic equation is shown on a white background. The equation reads '440 = k/26'. The number 440 is in light blue, and the number 26 is in red, while 'k' and '=' are black.", caption: none)], [Solve for the constant of variation.], [#figure(figph[A mathematical equation shows 26 multiplied by 440 equals 26 multiplied by the fraction k over 26.], alt: "A mathematical equation shows 26 multiplied by 440 equals 26 multiplied by the fraction k over 26.", caption: none)], [], [#figure(figph[A mathematical equation is presented, showing the value of k as 11,440. The text reads '11,440 = k' on a white background.], alt: "A mathematical equation is presented, showing the value of k as 11,440. The text reads '11,440 = k' on a white background.", caption: none)], [Write the equation that relates #math.equation(block: false, alt: "f")[$f$] and #math.equation(block: false, alt: "L")[$L$].], [#figure(figph[A mathematical equation, f = k/L, is displayed in black text on a white background, indicating that f is inversely proportional to L and directly proportional to k.], alt: "A mathematical equation, f = k/L, is displayed in black text on a white background, indicating that f is inversely proportional to L and directly proportional to k.", caption: none)], [Substitute in the constant of variation.], [#figure(figph[A mathematical equation shows f equals 11,440 divided by L. The variable 'f' is on the left side of the equation, and a fraction with '11,440' as the numerator and 'L' as the denominator is on the right.], alt: "A mathematical equation shows f equals 11,440 divided by L. The variable 'f' is on the left side of the equation, and a fraction with '11,440' as the numerator and 'L' as the denominator is on the right.", caption: none)], )) ⓑ #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([], [#math.equation(block: false, alt: "Find f when L equals 20 .")[$"Find" #h(0.2em) f #h(0.2em) "when" #h(0.2em) L = 20 .$]]), [Write the equation that relates #emph[f] and #emph[L].], [#math.equation(block: false, alt: "f equals the fraction 11,440 over L")[$f = frac(11","440, L)$]], [Substitute the given value for #emph[L].], [#math.equation(block: false, alt: "f equals the fraction 11,440 over 20")[$f = frac(11","440, 20)$]], [Simplify.], [#math.equation(block: false, alt: "f equals 572")[$f = 572$] #linebreak() A 20” guitar string has frequency #linebreak() 572 vibrations per second.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The number of hours it takes for ice to melt varies inversely with the air temperature. Suppose a block of ice melts in 2 hours when the temperature is 65 degrees. + ⓐ Write the equation of variation. + ⓑ How many hours would it take for the same block of ice to melt if the temperature was 78 degrees? #solutionbox[ ⓐ #math.equation(block: false, alt: "h equals the fraction 130 over t")[$h = frac(130, t)$] ⓑ #math.equation(block: false, alt: "1 the fraction 2 over 3")[$1 frac(2, 3)$] hours ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The force needed to break a board varies inversely with its length. Richard uses 24 pounds of pressure to break a 2-foot long board. + ⓐ Write the equation of variation. + ⓑ How many pounds of pressure is needed to break a 5-foot long board? #solutionbox[ ⓐ #math.equation(block: false, alt: "F equals the fraction 48 over L")[$F = frac(48, L)$] ⓑ 9.6 pounds ] ] === Section Exercises ==== Practice Makes Perfect #strong[Solve Direct Variation Problems] In the following exercises, solve. If #math.equation(block: false, alt: "y")[$y$] varies directly as #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y equals 14 , when x equals 3")[$y = 14 , #h(0.2em) "when" #h(0.2em) x = 3$], find the equation that relates #math.equation(block: false, alt: "x and y")[$x #h(0.2em) "and" #h(0.2em) y$]. #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 14 over 3 x")[$y = frac(14, 3) x$] ] If #math.equation(block: false, alt: "p")[$p$] varies directly as #math.equation(block: false, alt: "q")[$q$] and #math.equation(block: false, alt: "p equals 5 , when q equals 2")[$p = 5 , #h(0.2em) "when" #h(0.2em) q = 2$], find the equation that relates #math.equation(block: false, alt: "p and q")[$p #h(0.2em) "and" #h(0.2em) q$]. If #math.equation(block: false, alt: "v")[$v$] varies directly as #math.equation(block: false, alt: "w")[$w$] and #math.equation(block: false, alt: "v equals 24 , when w equals 8")[$v = 24 , #h(0.2em) "when" #h(0.2em) w = 8$], find the equation that relates #math.equation(block: false, alt: "v and w .")[$v #h(0.2em) "and" #h(0.2em) w .$] #solutionbox[ #math.equation(block: true, alt: "v equals 3 w")[$v = 3 w$] ] If #math.equation(block: false, alt: "a")[$a$] varies directly as #math.equation(block: false, alt: "b")[$b$] and #math.equation(block: false, alt: "a equals 16 , when b equals 4")[$a = 16 , #h(0.2em) "when" #h(0.2em) b = 4$], find the equation that relates #math.equation(block: false, alt: "a and b .")[$a #h(0.2em) "and" #h(0.2em) b .$] If #math.equation(block: false, alt: "p")[$p$] varies directly as #math.equation(block: false, alt: "q")[$q$] and #math.equation(block: false, alt: "p equals 9.6 , when q equals 3")[$p = 9.6 , #h(0.2em) "when" #h(0.2em) q = 3$], find the equation that relates #math.equation(block: false, alt: "p and q .")[$p #h(0.2em) "and" #h(0.2em) q .$] #solutionbox[ #math.equation(block: true, alt: "p equals 3.2 q")[$p = 3.2 q$] ] If #math.equation(block: false, alt: "y")[$y$] varies directly as #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y equals 12.4 , when x equals 4 ,")[$y = 12.4 , #h(0.2em) "when" #h(0.2em) x = 4 ,$] find the equation that relates #math.equation(block: false, alt: "x and y")[$x #h(0.2em) "and" #h(0.2em) y$] If #math.equation(block: false, alt: "a")[$a$] varies directly as #math.equation(block: false, alt: "b")[$b$] and #math.equation(block: false, alt: "a equals 6 , when b equals the fraction 1 over 3")[$a = 6 , #h(0.2em) "when" #h(0.2em) b = frac(1, 3)$], find the equation that relates #math.equation(block: false, alt: "a and b .")[$a #h(0.2em) "and" #h(0.2em) b .$] #solutionbox[ #math.equation(block: true, alt: "a equals 18 b")[$a = 18 b$] ] If #math.equation(block: false, alt: "v")[$v$] varies directly as #math.equation(block: false, alt: "w")[$w$] and #math.equation(block: false, alt: "v equals 8 , when w equals the fraction 1 over 2")[$v = 8 , #h(0.2em) "when" #h(0.2em) w = frac(1, 2)$], find the equation that relates #math.equation(block: false, alt: "v and w .")[$v #h(0.2em) "and" #h(0.2em) w .$] The amount of money Sally earns, #emph[P], varies directly with the number, #emph[n], of necklaces she sells. When Sally sells 15 necklaces she earns \$150. + ⓐ Write the equation that relates #emph[P] and #emph[n]. + ⓑ How much money would she earn if she sold 4 necklaces? #solutionbox[ ⓐ #math.equation(block: false, alt: "P equals 10 n")[$P = 10 n$] ⓑ #math.equation(block: false, alt: "$ 40")[$"$" 40$] ] The price, #emph[P], that Eric pays for gas varies directly with the number of gallons, #emph[g], he buys. It costs him \$50 to buy 20 gallons of gas. + ⓐ Write the equation that relates #emph[P] and #emph[g]. + ⓑ How much would 33 gallons cost Eric? Terri needs to make some pies for a fundraiser. The number of apples, #emph[a], varies directly with number of pies, #emph[p]. It takes nine apples to make two pies. + ⓐ Write the equation that relates #emph[a] and #emph[p]. + ⓑ How many apples would Terri need for six pies? #solutionbox[ ⓐ #math.equation(block: false, alt: "a equals 4.5 p")[$a = 4.5 p$] ⓑ 27 apples ] Joseph is traveling on a road trip. The distance, #emph[d], he travels before stopping for lunch varies directly with the speed, #emph[v], he travels. He can travel 120 miles at a speed of 60 mph. + ⓐ Write the equation that relates #emph[d] and #emph[v]. + ⓑ How far would he travel before stopping for lunch at a rate of 65 mph? The price of gas that Jesse purchased varies directly to how many gallons he purchased. He purchased 10 gallons of gas for \$39.80. + ⓐ Write the equation that relates the price to the number of gallons. + ⓑ How much will it cost Jesse for 15 gallons of gas? #solutionbox[ ⓐ #math.equation(block: false, alt: "p equals 3.98 g")[$p = 3.98 g$] ⓑ #math.equation(block: false, alt: "$ 59.70")[$"$" 59.70$] ] The distance that Sarah travels varies directly to how long she drives. She travels 440 miles in 8 hours. + ⓐ Write the equation that relates the distance to the number of hours. + ⓑ How far can Sally travel in 6 hours? The mass of a liquid varies directly with its volume. A liquid with mass 16 kilograms has a volume of 2 liters. + ⓐ Write the equation that relates the mass to the volume. + ⓑ What is the volume of this liquid if its mass is 128 kilograms? #solutionbox[ ⓐ #math.equation(block: false, alt: "m equals 8 v")[$m = 8 v$] ⓑ #math.equation(block: false, alt: "16 liters")[$"16 liters"$] ] The length that a spring stretches varies directly with a weight placed at the end of the spring. When Sarah placed a 10 pound watermelon on a hanging scale, the spring stretched 5 inches. + ⓐ Write the equation that relates the length of the spring to the weight. + ⓑ What weight of watermelon would stretch the spring 6 inches? The distance an object falls varies directly to the square of the time it falls. A ball falls 45 feet in 3 seconds. + ⓐ Write the equation that relates the distance to the time. + ⓑ How far will the ball fall in 7 seconds? #solutionbox[ ⓐ #math.equation(block: false, alt: "d equals 5 t squared")[$d = 5 t^(2)$] ⓑ 245 feet ] The maximum load a beam will support varies directly with the square of the diagonal of the beam’s cross-section. A beam with diagonal 6 inch will support a maximum load of 108 pounds. + ⓐ Write the equation that relates the load to the diagonal of the cross-section. + ⓑ What load will a beam with a 10 inch diagonal support? The area of a circle varies directly as the square of the radius. A circular pizza with a radius of 6 inches has an area of 113.04 square inches. + ⓐ Write the equation that relates the area to the radius. + ⓑ What is the area of a personal pizza with a radius 4 inches? #solutionbox[ ⓐ #math.equation(block: false, alt: "A equals 3.14 r squared")[$A = 3.14 r^(2)$] ⓑ #math.equation(block: false, alt: "50.24 sq. in.")[$50.24 #h(0.2em) "sq. in."$] ] The distance an object falls varies directly to the square of the time it falls. A ball falls 72 feet in 3 seconds, + ⓐ Write the equation that relates the distance to the time. + ⓑ How far will the ball have fallen in 8 seconds? #strong[Solve Inverse Variation Problems] In the following exercises, solve. If #math.equation(block: false, alt: "y")[$y$] varies inversely with #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y equals 5")[$y = 5$] when #math.equation(block: false, alt: "x equals 4")[$x = 4$] find the equation that relates #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y .")[$y .$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 20 over x")[$y = frac(20, x)$] ] If #math.equation(block: false, alt: "p")[$p$] varies inversely with #math.equation(block: false, alt: "q")[$q$] and #math.equation(block: false, alt: "p equals 2")[$p = 2$] when #math.equation(block: false, alt: "q equals 1")[$q = 1$] find the equation that relates #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "q .")[$q .$] If #math.equation(block: false, alt: "v")[$v$] varies inversely with #math.equation(block: false, alt: "w")[$w$] and #math.equation(block: false, alt: "v equals 6")[$v = 6$] when #math.equation(block: false, alt: "w equals the fraction 1 over 2")[$w = frac(1, 2)$] find the equation that relates #math.equation(block: false, alt: "v")[$v$] and #math.equation(block: false, alt: "w .")[$w .$] #solutionbox[ #math.equation(block: true, alt: "v equals the fraction 3 over w")[$v = frac(3, w)$] ] If #math.equation(block: false, alt: "a")[$a$] varies inversely with #math.equation(block: false, alt: "b")[$b$] and #math.equation(block: false, alt: "a equals 12")[$a = 12$] when #math.equation(block: false, alt: "b equals the fraction 1 over 3")[$b = frac(1, 3)$] find the equation that relates #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b .")[$b .$] Write an inverse variation equation to solve the following problems. The fuel consumption (mpg) of a car varies inversely with its weight. A Toyota Corolla weighs 2800 pounds and gets 33 mpg on the highway. + ⓐ Write the equation that relates the mpg to the car’s weight. + ⓑ What would the fuel consumption be for a Toyota Sequoia that weighs 5500 pounds? #solutionbox[ ⓐ #math.equation(block: false, alt: "g equals the fraction 92,400 over w")[$g = frac(92","400, w)$] ⓑ 16.8 mpg ] A car’s value varies inversely with its age. Jackie bought a 10 year old car for \$2,400. + ⓐ Write the equation that relates the car’s value to its age. + ⓑ What will be the value of Jackie’s car when it is 15 years old ? The time required to empty a tank varies inversely as the rate of pumping. It took Janet 5 hours to pump her flooded basement using a pump that was rated at 200 gpm (gallons per minute), + ⓐ Write the equation that relates the number of hours to the pump rate. + ⓑ How long would it take Janet to pump her basement if she used a pump rated at 400 gpm? #solutionbox[ ⓐ #math.equation(block: false, alt: "t equals the fraction 1000 over r")[$t = frac(1000, r)$] ⓑ #math.equation(block: false, alt: "2.5 hours")[$"2.5 hours"$] ] The volume of a gas in a container varies inversely as pressure on the gas. A container of helium has a volume of 370 cubic inches under a pressure of 15 psi. + ⓐ Write the equation that relates the volume to the pressure. + ⓑ What would be the volume of this gas if the pressure was increased to 20 psi? On a string instrument, the length of a string varies inversely as the frequency of its vibrations. An 11-inch string on a violin has a frequency of 400 cycles per second. + ⓐ Write the equation that relates the string length to its frequency. + ⓑ What is the frequency of a 10-inch string? #solutionbox[ ⓐ #math.equation(block: false, alt: "L equals the fraction 4,400 over f")[$L = frac(4","400, f)$] ⓑ 440 cycles per second ] Paul, a dentist, determined that the number of cavities that develops in his patient’s mouth each year varies inversely to the number of minutes spent brushing each night. His patient, Lori, had 4 cavities when brushing her teeth 30 seconds (0.5 minutes) each night. + ⓐ Write the equation that relates the number of cavities to the time spent brushing. + ⓑ How many cavities would Paul expect Lori to have if she had brushed her teeth for 2 minutes each night? The number of tickets for a sports fundraiser varies inversely to the price of each ticket. Brianna can buy 25 tickets at \$5each. + ⓐ Write the equation that relates the number of tickets to the price of each ticket. + ⓑ How many tickets could Brianna buy if the price of each ticket was \$2.50? #solutionbox[ ⓐ #math.equation(block: false, alt: "t equals the fraction 125 over p")[$t = frac(125, p)$] ⓑ 50 tickets ] Boyle’s Law states that if the temperature of a gas stays constant, then the pressure varies inversely to the volume of the gas. Braydon, a scuba diver, has a tank that holds 6 liters of air under a pressure of 220 psi. + ⓐ Write the equation that relates pressure to volume. + ⓑ If the pressure increases to 330 psi, how much air can Braydon’s tank hold? ==== Mixed Practice If #math.equation(block: false, alt: "y")[$y$] varies directly as #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y equals 5 , when x equals 3 .")[$y = 5 , #h(0.2em) "when" #h(0.2em) x = 3 .$], find the equation that relates #math.equation(block: false, alt: "x and y")[$x #h(0.2em) "and" #h(0.2em) y$]. #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 5 over 3 x")[$y = frac(5, 3) x$] ] If #math.equation(block: false, alt: "v")[$v$] varies directly as #math.equation(block: false, alt: "w")[$w$] and #math.equation(block: false, alt: "v equals 21 , when w equals 8 .")[$v = 21 , #h(0.2em) "when" #h(0.2em) w = 8 .$] find the equation that relates #math.equation(block: false, alt: "v and w .")[$v #h(0.2em) "and" #h(0.2em) w .$] If #math.equation(block: false, alt: "p")[$p$] varies inversely with #math.equation(block: false, alt: "q")[$q$] and #math.equation(block: false, alt: "p equals 5")[$p = 5$] when #math.equation(block: false, alt: "q equals 6")[$q = 6$], find the equation that relates #math.equation(block: false, alt: "p")[$p$] and #math.equation(block: false, alt: "q .")[$q .$] #solutionbox[ #math.equation(block: true, alt: "p equals the fraction 30 over q")[$p = frac(30, q)$] ] If #math.equation(block: false, alt: "y")[$y$] varies inversely with #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y equals 11")[$y = 11$] when #math.equation(block: false, alt: "x equals 3")[$x = 3$] find the equation that relates #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y .")[$y .$] If #math.equation(block: false, alt: "p")[$p$] varies directly as #math.equation(block: false, alt: "q")[$q$] and #math.equation(block: false, alt: "p equals 10 , when q equals 2 .")[$p = 10 , #h(0.2em) "when" #h(0.2em) q = 2 .$] find the equation that relates #math.equation(block: false, alt: "p and q")[$p #h(0.2em) "and" #h(0.2em) q$]. #solutionbox[ #math.equation(block: true, alt: "p equals 5 q")[$p = 5 q$] ] If #math.equation(block: false, alt: "v")[$v$] varies inversely with #math.equation(block: false, alt: "w")[$w$] and #math.equation(block: false, alt: "v equals 18")[$v = 18$] when #math.equation(block: false, alt: "w equals the fraction 1 over 3")[$w = frac(1, 3)$] find the equation that relates #math.equation(block: false, alt: "v")[$v$] and #math.equation(block: false, alt: "w .")[$w .$] The force needed to break a board varies inversely with its length. If Tom uses 20 pounds of pressure to break a 1.5-foot long board, how many pounds of pressure would he need to use to break a 6 foot long board? #solutionbox[ #math.equation(block: true, alt: "5 pounds")[$"5 pounds"$] ] The number of hours it takes for ice to melt varies inversely with the air temperature. A block of ice melts in 2.5 hours when the temperature is 54 degrees. How long would it take for the same block of ice to melt if the temperature was 45 degrees? The length a spring stretches varies directly with a weight placed at the end of the spring. When Meredith placed a 6-pound cantaloupe on a hanging scale, the spring stretched 2 inches. How far would the spring stretch if the cantaloupe weighed 9 pounds? #solutionbox[ #math.equation(block: true, alt: "3 inches")[$"3 inches"$] ] The amount that June gets paid varies directly the number of hours she works. When she worked 15 hours, she got paid \$111. How much will she be paid for working 18 hours? The fuel consumption (mpg) of a car varies inversely with its weight. A Ford Focus weighs 3000 pounds and gets 28.7 mpg on the highway. What would the fuel consumption be for a Ford Expedition that weighs 5,500 pounds? Round to the nearest tenth. #solutionbox[ #math.equation(block: true, alt: "15.6 mpg")[$15.6 #h(0.2em) "mpg"$] ] The volume of a gas in a container varies inversely as the pressure on the gas. If a container of argon has a volume of 336 cubic inches under a pressure of 2,500 psi, what will be its volume if the pressure is decreased to 2,000 psi? The distance an object falls varies directly to the square of the time it falls. If an object falls 52.8 feet in 4 seconds, how far will it fall in 9 seconds? #solutionbox[ #math.equation(block: true, alt: "267.3 feet")[$"267.3 feet"$] ] The area of the face of a Ferris wheel varies directly with the square of its radius. If the area of one face of a Ferris wheel with diameter 150 feet is 70,650 square feet, what is the area of one face of a Ferris wheel with diameter of 16 feet? ==== Everyday Math #strong[Ride Service] It costs \$35 for a ride from the city center to the airport, 14 miles away. + ⓐ Write the equation that relates the cost, #emph[c], with the number of miles, #emph[m]. + ⓑ What would it cost to travel 22 miles with this service? #solutionbox[ ⓐ #math.equation(block: false, alt: "c equals 2.5 m")[$c = 2.5 m$] ⓑ \$55 ] #strong[Road Trip] The number of hours it takes Jack to drive from Boston to Bangor is inversely proportional to his average driving speed. When he drives at an average speed of 40 miles per hour, it takes him 6 hours for the trip. + ⓐ Write the equation that relates the number of hours, #emph[h], with the speed, #emph[s.] + ⓑ How long would the trip take if his average speed was 75 miles per hour? ==== Writing Exercises In your own words, explain the difference between direct variation and inverse variation. #solutionbox[ Answers will vary. ] Make up an example from your life experience of inverse variation. ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[This image is four columns and three rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “solve direct variation problems.”, the next row reads “solve direct variation problems.” The remaining columns are blank.], alt: "This image is four columns and three rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads “solve direct variation problems.”, the next row reads “solve direct variation problems.” The remaining columns are blank.", caption: none) ⓑ After looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not? === Chapter 8 Review Exercises ==== Simplify Rational Expressions #strong[Determine the Values for Which a Rational Expression is Undefined] In the following exercises, determine the values for which the rational expression is undefined. #math.equation(block: true, alt: "the fraction 2 a plus 1 over 3 a minus 2")[$frac(2 a + 1, 3 a − 2)$] #solutionbox[ #math.equation(block: true, alt: "a not equal to the fraction 2 over 3")[$a ≠ frac(2, 3)$] ] #math.equation(block: true, alt: "the fraction b minus 3 over b squared minus 16")[$frac(b − 3, b^(2) − 16)$] #math.equation(block: true, alt: "the fraction 3 x y squared over 5 y")[$frac(3 x y^(2), 5 y)$] #solutionbox[ #math.equation(block: true, alt: "y not equal to 0")[$y ≠ 0$] ] #math.equation(block: true, alt: "the fraction u minus 3 over u squared minus u minus 30")[$frac(u − 3, u^(2) − u − 30)$] #strong[Evaluate Rational Expressions] In the following exercises, evaluate the rational expressions for the given values. #math.equation(block: true, alt: "the fraction 4 p minus 1 over p squared plus 5 when p equals −1")[$frac(4 p − 1, p^(2) + 5) #h(0.2em) "when" #h(0.2em) p = −1$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 5 over 6")[$− frac(5, 6)$] ] #math.equation(block: true, alt: "the fraction q squared minus 5 over q plus 3 when q equals 7")[$frac(q^(2) − 5, q + 3) #h(0.2em) "when q" = 7$] #math.equation(block: true, alt: "the fraction y squared minus 8 over y squared minus y minus 2 when y equals 1")[$frac(y^(2) − 8, y^(2) − y − 2) #h(0.2em) "when" #h(0.2em) y = 1$] #solutionbox[ #math.equation(block: true, alt: "the fraction 7 over 2")[$frac(7, 2)$] ] #math.equation(block: true, alt: "the fraction z squared plus 2 over 4 z minus z squared when z equals 3")[$frac(z^(2) + 2, 4 z − z^(2)) #h(0.2em) "when z" = 3$] #strong[Simplify Rational Expressions] In the following exercises, simplify. #math.equation(block: true, alt: "the fraction 10 over 24")[$frac(10, 24)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over 12")[$frac(5, 12)$] ] #math.equation(block: true, alt: "the fraction 8 m to the power 4 over 16 m n cubed")[$frac(8 m^(4), 16 m n^(3))$] #math.equation(block: true, alt: "the fraction 14 a minus 14 over a minus 1")[$frac(14 a − 14, a − 1)$] #solutionbox[ #math.equation(block: true, alt: "14")[$14$] ] #math.equation(block: true, alt: "the fraction b squared plus 7 b plus 12 over b squared plus 8 b plus 16")[$frac(b^(2) + 7 b + 12, b^(2) + 8 b + 16)$] #strong[Simplify Rational Expressions with Opposite Factors] In the following exercises, simplify. #math.equation(block: true, alt: "the fraction c squared minus c minus 2 over 4 minus c squared")[$frac(c^(2) − c − 2, 4 − c^(2))$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction c plus 1 over c plus 2")[$− frac(c + 1, c + 2)$] ] #math.equation(block: true, alt: "the fraction d minus 16 over 16 minus d")[$frac(d − 16, 16 − d)$] #math.equation(block: true, alt: "the fraction 7 v minus 35 over 25 minus v squared")[$frac(7 v − 35, 25 − v^(2))$] #solutionbox[ #math.equation(block: true, alt: "minus the fraction 7 over 5 plus v")[$− frac(7, 5 + v)$] ] #math.equation(block: true, alt: "the fraction w squared minus 3 w minus 28 over 49 minus w squared")[$frac(w^(2) − 3 w − 28, 49 − w^(2))$] ==== Multiply and Divide Rational Expressions #strong[Multiply Rational Expressions] In the following exercises, multiply. #math.equation(block: true, alt: "the fraction 3 over 8 times the fraction 2 over 15")[$frac(3, 8) · frac(2, 15)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 20")[$frac(1, 20)$] ] #math.equation(block: true, alt: "the fraction 2 x y squared over 8 y cubed times the fraction 16 y over 24 x")[$frac(2 x y^(2), 8 y^(3)) · frac(16 y, 24 x)$] #math.equation(block: true, alt: "the fraction 3 a squared plus 21 a over a squared plus 6 a minus 7 times the fraction a minus 1 over a b")[$frac(3 a^(2) + 21 a, a^(2) + 6 a − 7) · frac(a − 1, a b)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over b")[$frac(3, b)$] ] #math.equation(block: true, alt: "the fraction 5 z squared over 5 z squared plus 40 z plus 35 times the fraction z squared minus 1 over 3 z")[$frac(5 z^(2), 5 z^(2) + 40 z + 35) · frac(z^(2) − 1, 3 z)$] #strong[Divide Rational Expressions] In the following exercises, divide. #math.equation(block: true, alt: "the fraction t squared minus 4 t minus 12 over t squared plus 8 t plus 12 divided by the fraction t squared minus 36 over 6 t")[$frac(t^(2) − 4 t − 12, t^(2) + 8 t + 12) ÷ frac(t^(2) − 36, 6 t)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 6 t over open parenthesis t plus 6 close parenthesis squared")[$frac(6 t, ( t + 6 )^(2))$] ] #math.equation(block: true, alt: "the fraction r squared minus 16 over 4 divided by the fraction r cubed minus 64 over 2 r squared plus 8 r plus 32")[$frac(r^(2) − 16, 4) ÷ frac(r^(3) − 64, 2 r^(2) + 8 r + 32)$] #math.equation(block: true, alt: "the fraction 11 plus w over w minus 9 divided by the fraction 121 minus w squared over 9 minus w")[$frac(11 + w, w − 9) ÷ frac(121 − w^(2), 9 − w)$] #solutionbox[ #math.equation(block: true, alt: "the fraction −1 over 11 minus w")[$frac(−1, 11 − w)$] ] #math.equation(block: true, alt: "the fraction 3 y squared minus 12 y minus 63 over 4 y plus 3 divided by open parenthesis 6 y squared minus 42 y close parenthesis")[$frac(3 y^(2) − 12 y − 63, 4 y + 3) ÷ ( 6 y^(2) − 42 y )$] #math.equation(block: true, alt: "the fraction the fraction c squared minus 64 over 3 c squared plus 26 c plus 16 over the fraction c squared minus 4 c minus 32 over 15 c plus 10")[$frac(frac(c^(2) − 64, 3 c^(2) + 26 c + 16), frac(c^(2) − 4 c − 32, 15 c + 10))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 over c plus 4")[$frac(5, c + 4)$] ] #math.equation(block: true, alt: "the fraction 8 m squared minus 8 m over m minus 4 times the fraction m squared plus 2 m minus 24 over m squared plus 7 m plus 10 divided by the fraction 2 m squared minus 6 m over m plus 5")[$frac(8 m^(2) − 8 m, m − 4) · frac(m^(2) + 2 m − 24, m^(2) + 7 m + 10) ÷ frac(2 m^(2) − 6 m, m + 5)$] ==== Add and Subtract Rational Expressions with a Common Denominator #strong[Add Rational Expressions with a Common Denominator] In the following exercises, add. #math.equation(block: true, alt: "the fraction 3 over 5 plus the fraction 2 over 5")[$frac(3, 5) + frac(2, 5)$] #solutionbox[ #math.equation(block: true, alt: "1")[$1$] ] #math.equation(block: true, alt: "the fraction 4 a squared over 2 a minus 1 minus the fraction 1 over 2 a minus 1")[$frac(4 a^(2), 2 a − 1) − frac(1, 2 a − 1)$] #math.equation(block: true, alt: "the fraction p squared plus 10 p over p plus 5 plus the fraction 25 over p plus 5")[$frac(p^(2) + 10 p, p + 5) + frac(25, p + 5)$] #solutionbox[ #math.equation(block: true, alt: "p plus 5")[$p + 5$] ] #math.equation(block: true, alt: "the fraction 3 x over x minus 1 plus the fraction 2 over x minus 1")[$frac(3 x, x − 1) + frac(2, x − 1)$] #strong[Subtract Rational Expressions with a Common Denominator] In the following exercises, subtract. #math.equation(block: true, alt: "the fraction d squared over d plus 4 minus the fraction 3 d plus 28 over d plus 4")[$frac(d^(2), d + 4) − frac(3 d + 28, d + 4)$] #solutionbox[ #math.equation(block: true, alt: "d minus 7")[$d − 7$] ] #math.equation(block: true, alt: "the fraction z squared over z plus 10 minus the fraction 100 over z plus 10")[$frac(z^(2), z + 10) − frac(100, z + 10)$] #math.equation(block: true, alt: "the fraction 4 q squared minus q plus 3 over q squared plus 6 q plus 5 minus the fraction 3 q squared plus q plus 6 over q squared plus 6 q plus 5")[$frac(4 q^(2) − q + 3, q^(2) + 6 q + 5) − frac(3 q^(2) + q + 6, q^(2) + 6 q + 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction q minus 3 over q plus 5")[$frac(q − 3, q + 5)$] ] #math.equation(block: true, alt: "the fraction 5 t squared plus 4 t plus 3 over t squared minus 25 minus the fraction 4 t squared minus 8 t minus 32 over t squared minus 25")[$frac(5 t^(2) + 4 t + 3, t^(2) − 25) − frac(4 t^(2) − 8 t − 32, t^(2) − 25)$] #strong[Add and Subtract Rational Expressions whose Denominators are Opposites] In the following exercises, add and subtract. #math.equation(block: true, alt: "the fraction 18 w over 6 w minus 1 plus the fraction 3 w minus 2 over 1 minus 6 w")[$frac(18 w, 6 w − 1) + frac(3 w − 2, 1 − 6 w)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 15 w plus 2 over 6 w minus 1")[$frac(15 w + 2, 6 w − 1)$] ] #math.equation(block: true, alt: "the fraction a squared plus 3 a over a squared minus 16 minus the fraction 3 a plus 8 over 16 minus a squared")[$frac(a^(2) + 3 a, a^(2) − 16) − frac(3 a + 8, 16 − a^(2))$] #math.equation(block: true, alt: "the fraction 2 b squared plus 3 b minus 15 over b squared minus 49 minus the fraction b squared plus 16 b minus 1 over 49 minus b squared")[$frac(2 b^(2) + 3 b − 15, b^(2) − 49) − frac(b^(2) + 16 b − 1, 49 − b^(2))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 b squared plus 19 b minus 16 over b minus 7 b plus 7")[$frac(3 b^(2) + 19 b − 16, b − 7 b + 7)$] ] #math.equation(block: true, alt: "the fraction 8 y squared minus 10 y plus 7 over 2 y minus 5 plus the fraction 2 y squared plus 7 y plus 2 over 5 minus 2 y")[$frac(8 y^(2) − 10 y + 7, 2 y − 5) + frac(2 y^(2) + 7 y + 2, 5 − 2 y)$] ==== Add and Subtract Rational Expressions With Unlike Denominators #strong[Find the Least Common Denominator of Rational Expressions] In the following exercises, find the LCD. #math.equation(block: true, alt: "the fraction 4 over m squared minus 3 m minus 10 , the fraction 2 m over m squared minus m minus 20")[$frac(4, m^(2) − 3 m − 10) , frac(2 m, m^(2) − m − 20)$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis m plus 2 close parenthesis open parenthesis m minus 5 close parenthesis open parenthesis m plus 4 close parenthesis")[$( m + 2 ) ( m − 5 ) ( m + 4 )$] ] #math.equation(block: true, alt: "the fraction 6 over n squared minus 4 , the fraction 2 n over n squared minus 4 n plus 4")[$frac(6, n^(2) − 4) , frac(2 n, n^(2) − 4 n + 4)$] #math.equation(block: true, alt: "the fraction 5 over 3 p squared plus 19 p plus 6 , the fraction 2 p over 3 p squared plus 25 p plus 8")[$frac(5, 3 p^(2) + 19 p + 6) , frac(2 p, 3 p^(2) + 25 p + 8)$] #solutionbox[ #math.equation(block: true, alt: "open parenthesis 3 p plus 1 close parenthesis open parenthesis p plus 6 close parenthesis open parenthesis p plus 8 close parenthesis")[$( 3 p + 1 ) ( p + 6 ) ( p + 8 )$] ] #strong[Find Equivalent Rational Expressions] In the following exercises, rewrite as equivalent rational expressions with the given denominator. Rewrite as equivalent rational expressions with denominator #math.equation(block: false, alt: "open parenthesis m plus 2 close parenthesis open parenthesis m minus 5 close parenthesis open parenthesis m plus 4 close parenthesis")[$( m + 2 ) ( m − 5 ) ( m + 4 )$]: #math.equation(block: true, alt: "the fraction 4 over m squared minus 3 m minus 10 , the fraction 2 m over m squared minus m minus 20 .")[$frac(4, m^(2) − 3 m − 10) , frac(2 m, m^(2) − m − 20) .$]Rewrite as equivalent rational expressions with denominator #math.equation(block: false, alt: "open parenthesis n minus 2 close parenthesis open parenthesis n minus 2 close parenthesis open parenthesis n plus 2 close parenthesis")[$( n − 2 ) ( n − 2 ) ( n + 2 )$]: #math.equation(block: true, alt: "the fraction 6 over n squared minus 4 n plus 4 , the fraction 2 n over n squared minus 4 .")[$frac(6, n^(2) − 4 n + 4) , frac(2 n, n^(2) − 4) .$]#solutionbox[ #math.equation(block: true, alt: "the fraction 6 n plus 12 over open parenthesis n minus 2 close parenthesis open parenthesis n minus 2 close parenthesis open parenthesis n plus 2 close parenthesis ,")[$frac(6 n + 12, ( n − 2 ) ( n − 2 ) ( n + 2 )) ,$] #linebreak() #math.equation(block: true, alt: "the fraction 2 n squared minus 4 n over open parenthesis n minus 2 close parenthesis open parenthesis n minus 2 close parenthesis open parenthesis n plus 2 close parenthesis")[$frac(2 n^(2) − 4 n, ( n − 2 ) ( n − 2 ) ( n + 2 ))$] ] Rewrite as equivalent rational expressions with denominator #math.equation(block: false, alt: "open parenthesis 3 p plus 1 close parenthesis open parenthesis p plus 6 close parenthesis open parenthesis p plus 8 close parenthesis")[$( 3 p + 1 ) ( p + 6 ) ( p + 8 )$]: #math.equation(block: true, alt: "the fraction 5 over 3 p squared plus 19 p plus 6 , the fraction 7 p over 3 p squared plus 25 p plus 8")[$frac(5, 3 p^(2) + 19 p + 6) , frac(7 p, 3 p^(2) + 25 p + 8)$]#strong[Add Rational Expressions with Different Denominators] In the following exercises, add. #math.equation(block: true, alt: "the fraction 2 over 3 plus the fraction 3 over 5")[$frac(2, 3) + frac(3, 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 19 over 15")[$frac(19, 15)$] ] #math.equation(block: true, alt: "the fraction 7 over 5 a plus the fraction 3 over 2 b")[$frac(7, 5 a) + frac(3, 2 b)$] #math.equation(block: true, alt: "the fraction 2 over c minus 2 plus the fraction 9 over c plus 3")[$frac(2, c − 2) + frac(9, c + 3)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 11 c minus 12 over open parenthesis c minus 2 close parenthesis open parenthesis c plus 3 close parenthesis")[$frac(11 c − 12, ( c − 2 ) ( c + 3 ))$] ] #math.equation(block: true, alt: "the fraction 3 d over d squared minus 9 plus the fraction 5 over d squared plus 6 d plus 9")[$frac(3 d, d^(2) − 9) + frac(5, d^(2) + 6 d + 9)$] #math.equation(block: true, alt: "the fraction 2 x over x squared plus 10 x plus 24 plus the fraction 3 x over x squared plus 8 x plus 16")[$frac(2 x, x^(2) + 10 x + 24) + frac(3 x, x^(2) + 8 x + 16)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 5 x squared plus 26 x over open parenthesis x plus 4 close parenthesis open parenthesis x plus 4 close parenthesis open parenthesis x plus 6 close parenthesis")[$frac(5 x^(2) + 26 x, ( x + 4 ) ( x + 4 ) ( x + 6 ))$] ] #math.equation(block: true, alt: "the fraction 5 q over p squared q minus p squared plus the fraction 4 q over q squared minus 1")[$frac(5 q, p^(2) q − p^(2)) + frac(4 q, q^(2) − 1)$] #strong[Subtract Rational Expressions with Different Denominators] In the following exercises, subtract and add. #math.equation(block: true, alt: "the fraction 3 v over v plus 2 minus the fraction v plus 2 over v plus 8")[$frac(3 v, v + 2) − frac(v + 2, v + 8)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 open parenthesis v squared plus 10 v minus 2 close parenthesis over open parenthesis v plus 2 close parenthesis open parenthesis v plus 8 close parenthesis")[$frac(2 ( v^(2) + 10 v − 2 ), ( v + 2 ) ( v + 8 ))$] ] #math.equation(block: true, alt: "the fraction −3 w minus 15 over w squared plus w minus 20 minus the fraction w plus 2 over 4 minus w")[$frac(−3 w − 15, w^(2) + w − 20) − frac(w + 2, 4 − w)$] #math.equation(block: true, alt: "the fraction 7 m plus 3 over m plus 2 minus 5")[$frac(7 m + 3, m + 2) − 5$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 m minus 7 over m plus 2")[$frac(2 m − 7, m + 2)$] ] #math.equation(block: true, alt: "the fraction n over n plus 3 plus the fraction 2 over n minus 3 minus the fraction n minus 9 over n squared minus 9")[$frac(n, n + 3) + frac(2, n − 3) − frac(n − 9, n^(2) − 9)$] #math.equation(block: true, alt: "the fraction 8 d over d squared minus 64 minus the fraction 4 over d plus 8")[$frac(8 d, d^(2) − 64) − frac(4, d + 8)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 over d minus 8")[$frac(4, d − 8)$] ] #math.equation(block: true, alt: "the fraction 5 over 12 x squared y plus the fraction 7 over 20 x y cubed")[$frac(5, 12 x^(2) y) + frac(7, 20 x y^(3))$] ==== Simplify Complex Rational Expressions #strong[Simplify a Complex Rational Expression by Writing it as Division] In the following exercises, simplify. #math.equation(block: true, alt: "the fraction the fraction 5 a over a plus 2 over the fraction 10 a squared over a squared minus 4")[$frac(frac(5 a, a + 2), frac(10 a^(2), a^(2) − 4))$] #solutionbox[ #math.equation(block: true, alt: "the fraction a minus 2 over 2 a")[$frac(a − 2, 2 a)$] ] #math.equation(block: true, alt: "the fraction the fraction 2 over 5 plus the fraction 5 over 6 over the fraction 1 over 3 plus the fraction 1 over 4")[$frac(frac(2, 5) + frac(5, 6), frac(1, 3) + frac(1, 4))$] #math.equation(block: true, alt: "the fraction x minus the fraction 3 x over x plus 5 over the fraction 1 over x plus 5 plus the fraction 1 over x minus 5")[$frac(x − frac(3 x, x + 5), frac(1, x + 5) + frac(1, x − 5))$] #solutionbox[ #math.equation(block: true, alt: "the fraction open parenthesis x plus 2 close parenthesis open parenthesis x minus 5 close parenthesis over 2")[$frac(( x + 2 ) ( x − 5 ), 2)$] ] #math.equation(block: true, alt: "the fraction the fraction 2 over m plus the fraction m over n over the fraction n over m minus the fraction 1 over n")[$frac(frac(2, m) + frac(m, n), frac(n, m) − frac(1, n))$] #strong[Simplify a Complex Rational Expression by Using the LCD] In the following exercises, simplify. #math.equation(block: true, alt: "the fraction 6 plus the fraction 2 over q minus 4 over the fraction 5 over q plus 4")[$frac(6 + frac(2, q − 4), frac(5, q + 4))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 2 3 q minus 11 q plus 4 over 5 open parenthesis q minus 4 close parenthesis")[$frac(2 3 q − 11 q + 4, 5 ( q − 4 ))$] ] #math.equation(block: true, alt: "the fraction the fraction 3 over a squared minus the fraction 1 over b over the fraction 1 over a plus the fraction 1 over b squared")[$frac(frac(3, a^(2)) − frac(1, b), frac(1, a) + frac(1, b^(2)))$] #math.equation(block: true, alt: "the fraction the fraction 2 over z squared minus 49 plus the fraction 1 over z plus 7 over the fraction 9 over z plus 7 plus the fraction 12 over z minus 7")[$frac(frac(2, z^(2) − 49) + frac(1, z + 7), frac(9, z + 7) + frac(12, z − 7))$] #solutionbox[ #math.equation(block: true, alt: "the fraction z minus 5 over 21 z plus 21")[$frac(z − 5, 21 z + 21)$] ] #math.equation(block: true, alt: "the fraction the fraction 3 over y squared minus 4 y minus 32 over the fraction 2 over y minus 8 plus the fraction 1 over y plus 4")[$frac(frac(3, y^(2) − 4 y − 32), frac(2, y − 8) + frac(1, y + 4))$] ==== Solve Rational Equations #strong[Solve Rational Equations] In the following exercises, solve. #math.equation(block: true, alt: "the fraction 1 over 2 plus the fraction 2 over 3 equals the fraction 1 over x")[$frac(1, 2) + frac(2, 3) = frac(1, x)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 6 over 7")[$frac(6, 7)$] ] #math.equation(block: true, alt: "1 minus the fraction 2 over m equals the fraction 8 over m squared")[$1 − frac(2, m) = frac(8, m^(2))$] #math.equation(block: true, alt: "the fraction 1 over b minus 2 plus the fraction 1 over b plus 2 equals the fraction 3 over b squared minus 4")[$frac(1, b − 2) + frac(1, b + 2) = frac(3, b^(2) − 4)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 3 over 2")[$frac(3, 2)$] ] #math.equation(block: true, alt: "the fraction 3 over q plus 8 minus the fraction 2 over q minus 2 equals 1")[$frac(3, q + 8) − frac(2, q − 2) = 1$] #math.equation(block: true, alt: "the fraction v minus 15 over v squared minus 9 v plus 18 equals the fraction 4 over v minus 3 plus the fraction 2 over v minus 6")[$frac(v − 15, v^(2) − 9 v + 18) = frac(4, v − 3) + frac(2, v − 6)$] #solutionbox[ #math.equation(block: true, alt: "no solution")[$"no solution"$] ] #math.equation(block: true, alt: "the fraction z over 12 plus the fraction z plus 3 over 3 z equals the fraction 1 over z")[$frac(z, 12) + frac(z + 3, 3 z) = frac(1, z)$] #strong[Solve a Rational Equation for a Specific Variable] In the following exercises, solve for the indicated variable. #math.equation(block: true, alt: "the fraction V over l equals h w for l")[$frac(V, l) = h w #h(0.2em) "for" #h(0.2em) l$] #solutionbox[ #math.equation(block: true, alt: "l equals the fraction V over h w")[$l = frac(V, h w)$] ] #math.equation(block: true, alt: "the fraction 1 over x minus the fraction 2 over y equals 5 for y")[$frac(1, x) − frac(2, y) = 5 #h(0.2em) "for" #h(0.2em) y$] #math.equation(block: true, alt: "x equals the fraction y plus 5 over z minus 7 for z")[$x = frac(y + 5, z − 7) #h(0.2em) "for" #h(0.2em) z$] #solutionbox[ #math.equation(block: true, alt: "z equals the fraction y plus 5 plus 7 x over x")[$z = frac(y + 5 + 7 x, x)$] ] #math.equation(block: true, alt: "P equals the fraction k over V for V")[$P = frac(k, V) #h(0.2em) "for" #h(0.2em) V$] ==== Solve Proportion and Similar Figure Applications Similarity #strong[Solve Proportions] In the following exercises, solve. #math.equation(block: true, alt: "the fraction x over 4 equals the fraction 3 over 5")[$frac(x, 4) = frac(3, 5)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 12 over 5")[$frac(12, 5)$] ] #math.equation(block: true, alt: "the fraction 3 over y equals the fraction 9 over 5")[$frac(3, y) = frac(9, 5)$] #math.equation(block: true, alt: "the fraction s over s plus 20 equals the fraction 3 over 7")[$frac(s, s + 20) = frac(3, 7)$] #solutionbox[ #math.equation(block: true, alt: "15")[$15$] ] #math.equation(block: true, alt: "the fraction t minus 3 over 5 equals the fraction t plus 2 over 9")[$frac(t − 3, 5) = frac(t + 2, 9)$] In the following exercises, solve using proportions. Rachael had a 21 ounce strawberry shake that has 739 calories. How many calories are there in a 32 ounce shake? #solutionbox[ #math.equation(block: true, alt: "1126 calories")[$1126 #h(0.2em) "calories"$] ] Leo went to Mexico over Christmas break and changed \$525 dollars into Mexican pesos. At that time, the exchange rate had \$1 US is equal to 16.25 Mexican pesos. How many Mexican pesos did he get for his trip? #strong[Solve Similar Figure Applications] In the following exercises, solve. ∆ABC is similar to ∆XYZ. The lengths of two sides of each triangle are given in the figure. Find the lengths of the third sides. #figure(figph[This image shows two triangles. The large triangle is labeled A B C. The length from A to B is labeled 8. The length from B to C is labeled 7. The length from C to A is labeled b. The smaller triangle is triangle x y z. The length from x to y is labeled 2 and two-thirds. The length from y to z is labeled x. The length from x to z is labeled 3.], alt: "This image shows two triangles. The large triangle is labeled A B C. The length from A to B is labeled 8. The length from B to C is labeled 7. The length from C to A is labeled b. The smaller triangle is triangle x y z. The length from x to y is labeled 2 and two-thirds. The length from y to z is labeled x. The length from x to z is labeled 3.", caption: none) #solutionbox[ #math.equation(block: true, alt: "b equals 9 ; x equals 2 the fraction 1 over 3")[$b = 9 ; x = 2 frac(1, 3)$] ] On a map of Europe, Paris, Rome, and Vienna form a triangle whose sides are shown in the figure below. If the actual distance from Rome to Vienna is 700 miles, find the distance from + ⓐ Paris to Rome + ⓑ Paris to Vienna #figure(figph[This is an image of a triangle. Clockwise beginning at the top, each vertex is labeled. The top vertex is labeled “Paris”, the next vertex is labeled “Vienna”, and the next vertex is labeled “Rome”. The distance from Paris to Vienna is 7.7 centimeters. The distance from Vienna to Rome is 7 centimeters. The distance from Rome to Paris is 8.9 centimeters.], alt: "This is an image of a triangle. Clockwise beginning at the top, each vertex is labeled. The top vertex is labeled “Paris”, the next vertex is labeled “Vienna”, and the next vertex is labeled “Rome”. The distance from Paris to Vienna is 7.7 centimeters. The distance from Vienna to Rome is 7 centimeters. The distance from Rome to Paris is 8.9 centimeters.", caption: none) Tony is 5.75 feet tall. Late one afternoon, his shadow was 8 feet long. At the same time, the shadow of a nearby tree was 32 feet long. Find the height of the tree. #solutionbox[ 23 feet ] The height of a lighthouse in Pensacola, Florida is 150 feet. Standing next to the statue, 5.5 foot tall Natalie cast a 1.1 foot shadow How long would the shadow of the lighthouse be? ==== Solve Uniform Motion and Work Applications Problems #strong[Solve Uniform Motion Applications] In the following exercises, solve. When making the 5-hour drive home from visiting her parents, Lisa ran into bad weather. She was able to drive 176 miles while the weather was good, but then driving 10 mph slower, went 81 miles in the bad weather. How fast did she drive when the weather was bad? #solutionbox[ #math.equation(block: true, alt: "45 mph")[$"45 mph"$] ] Mark is riding on a plane that can fly 490 miles with a headwind of 20 mph in the same time that it can fly 350 miles against a tailwind of 20 mph. What is the speed of the plane? John can ride his bicycle 8 mph faster than Luke can ride his bike. It takes Luke 3 hours longer than John to ride 48 miles. How fast can John ride his bike? #solutionbox[ #math.equation(block: true, alt: "16 mph")[$"16 mph"$] ] Mark was training for a triathlon. He ran 8 kilometers and biked 32 kilometers in a total of 3 hours. His running speed was 8 kilometers per hour less than his biking speed. What was his running speed? #strong[Solve Work Applications] In the following exercises, solve. Jerry can frame a room in 1 hour, while Jake takes 4 hours. How long could they frame a room working together? #solutionbox[ #math.equation(block: true, alt: "the fraction 4 over 5 hour")[$frac(4, 5) #h(0.2em) "hour"$] ] Lisa takes 3 hours to mow the lawn while her cousin, Barb, takes 2 hours. How long will it take them working together? Jeffrey can paint a house in 6 days, but if he gets a helper he can do it in 4 days. How long would it take the helper to paint the house alone? #solutionbox[ #math.equation(block: true, alt: "12 days")[$12 #h(0.2em) "days"$] ] Sue and Deb work together writing a book that takes them 90 days. If Sue worked alone it would take her 120 days. How long would it take Deb to write the book alone? ==== Use Direct and Inverse Variation #strong[Solve Direct Variation Problems] In the following exercises, solve. If #math.equation(block: false, alt: "y")[$y$] varies directly as #math.equation(block: false, alt: "x")[$x$], when #math.equation(block: false, alt: "y equals 9")[$y = 9$] and #math.equation(block: false, alt: "x equals 3")[$x = 3$], find #math.equation(block: false, alt: "x")[$x$] when #math.equation(block: false, alt: "y equals 21 .")[$y = 21 .$] #solutionbox[ #math.equation(block: true, alt: "7")[$7$] ] If #math.equation(block: false, alt: "y")[$y$] varies inversely as #math.equation(block: false, alt: "x")[$x$], when #math.equation(block: false, alt: "y equals 20")[$y = 20$] and #math.equation(block: false, alt: "x equals 2")[$x = 2$] find #math.equation(block: false, alt: "y")[$y$] when #math.equation(block: false, alt: "x equals 4 .")[$x = 4 .$] If #math.equation(block: false, alt: "m")[$m$] varies inversely with the square of #math.equation(block: false, alt: "n")[$n$], when #math.equation(block: false, alt: "m equals 4")[$m = 4$] and #math.equation(block: false, alt: "n equals 6")[$n = 6$] find #math.equation(block: false, alt: "m")[$m$] when #math.equation(block: false, alt: "n equals 2 .")[$n = 2 .$] #solutionbox[ #math.equation(block: true, alt: "36")[$36$] ] Vanessa is traveling to see her fiancé. The distance, #emph[d], varies directly with the speed, #emph[v], she drives. If she travels 258 miles driving 60 mph, how far would she travel going 70 mph? If the cost of a pizza varies directly with its diameter, and if an 8” diameter pizza costs \$12, how much would a 6” diameter pizza cost? #solutionbox[ #math.equation(block: true, alt: "$ 9")[$"$" 9$] ] The distance to stop a car varies directly with the square of its speed. It takes 200 feet to stop a car going 50 mph. How many feet would it take to stop a car going 60 mph? #strong[Solve Inverse Variation Problems] In the following exercises, solve. The number of tickets for a music fundraiser varies inversely with the price of the tickets. If Madelyn has just enough money to purchase 12 tickets for \$6 each, how many tickets can Madelyn afford to buy if the price increased to \$8? #solutionbox[ #math.equation(block: true, alt: "9 tickets")[$9 #h(0.2em) "tickets"$] ] On a string instrument, the length of a string varies inversely with the frequency of its vibrations. If an 11-inch string on a violin has a frequency of 360 cycles per second, what frequency does a 12 inch string have? === Practice Test In the following exercises, simplify. #math.equation(block: true, alt: "the fraction 3 a squared b over 6 a b squared")[$frac(3 a^(2) b, 6 a b^(2))$] #solutionbox[ #math.equation(block: true, alt: "the fraction a over 2 b")[$frac(a, 2 b)$] ] #math.equation(block: true, alt: "the fraction 5 b minus 25 over b squared minus 25")[$frac(5 b − 25, b^(2) − 25)$] In the following exercises, perform the indicated operation and simplify. #math.equation(block: true, alt: "the fraction 4 x over x plus 2 times the fraction x squared plus 5 x plus 6 over 12 x squared")[$frac(4 x, x + 2) · frac(x^(2) + 5 x + 6, 12 x^(2))$] #solutionbox[ #math.equation(block: true, alt: "the fraction x plus 3 over 3 x")[$frac(x + 3, 3 x)$] ] #math.equation(block: true, alt: "the fraction 5 y over 4 y minus 8 times the fraction y squared minus 4 over 10")[$frac(5 y, 4 y − 8) · frac(y^(2) − 4, 10)$] #math.equation(block: true, alt: "the fraction 4 over p q plus the fraction 5 over p")[$frac(4, p q) + frac(5, p)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 4 plus 5 q over p q")[$frac(4 + 5 q, p q)$] ] #math.equation(block: true, alt: "the fraction 1 over z minus 9 minus the fraction 3 over z plus 9")[$frac(1, z − 9) − frac(3, z + 9)$] #math.equation(block: true, alt: "the fraction the fraction 2 over 3 plus the fraction 3 over 5 over the fraction 2 over 5")[$frac(frac(2, 3) + frac(3, 5), frac(2, 5))$] #solutionbox[ #math.equation(block: true, alt: "the fraction 19 over 6")[$frac(19, 6)$] ] #math.equation(block: true, alt: "the fraction the fraction 1 over m minus the fraction 1 over n over the fraction 1 over n plus the fraction 1 over m")[$frac(frac(1, m) − frac(1, n), frac(1, n) + frac(1, m))$] In the following exercises, solve each equation. #math.equation(block: true, alt: "the fraction 1 over 2 plus the fraction 2 over 7 equals the fraction 1 over x")[$frac(1, 2) + frac(2, 7) = frac(1, x)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 14 over 11")[$frac(14, 11)$] ] #math.equation(block: true, alt: "the fraction 5 over y minus 6 equals the fraction 3 over y plus 6")[$frac(5, y − 6) = frac(3, y + 6)$] #math.equation(block: true, alt: "the fraction 1 over z minus 5 plus the fraction 1 over z plus 5 equals the fraction 1 over z squared minus 25")[$frac(1, z − 5) + frac(1, z + 5) = frac(1, z^(2) − 25)$] #solutionbox[ #math.equation(block: true, alt: "the fraction 1 over 2")[$frac(1, 2)$] ] #math.equation(block: true, alt: "the fraction t over 4 equals the fraction 3 over 5")[$frac(t, 4) = frac(3, 5)$] #math.equation(block: true, alt: "the fraction 2 over r minus 2 equals the fraction 3 over r minus 1")[$frac(2, r − 2) = frac(3, r − 1)$] #solutionbox[ 4 ] In the following exercises, solve. If #math.equation(block: false, alt: "y")[$y$] varies directly with #math.equation(block: false, alt: "x ,")[$x ,$] and #math.equation(block: false, alt: "x equals 5")[$x = 5$] when #math.equation(block: false, alt: "y equals 30 ,")[$y = 30 ,$] find #math.equation(block: false, alt: "x")[$x$] when #math.equation(block: false, alt: "y equals 42 .")[$y = 42 .$] If #math.equation(block: false, alt: "y")[$y$] varies inversely with #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "x equals 6")[$x = 6$] when #math.equation(block: false, alt: "y equals 20 ,")[$y = 20 ,$] find #math.equation(block: false, alt: "y")[$y$] when #math.equation(block: false, alt: "x equals 2 .")[$x = 2 .$] #solutionbox[ 60 ] If #math.equation(block: false, alt: "y")[$y$] varies inversely with the square of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "x equals 3")[$x = 3$] when #math.equation(block: false, alt: "y equals 9 ,")[$y = 9 ,$] find #math.equation(block: false, alt: "y")[$y$] when #math.equation(block: false, alt: "x equals 4 .")[$x = 4 .$] The recommended erythromycin dosage for dogs, is 5 mg for every pound the dog weighs. If Daisy weighs 25 pounds, how many milligrams of erythromycin should her veterinarian prescribe? #solutionbox[ #math.equation(block: true, alt: "125 mg")[$125 #h(0.2em) "mg"$] ] Julia spent 4 hours Sunday afternoon exercising at the gym. She ran on the treadmill for 10 miles and then biked for 20 miles. Her biking speed was 5 mph faster than her running speed on the treadmill. What was her running speed? Kurt can ride his bike for 30 miles with the wind in the same amount of time that he can go 21 miles against the wind. If the wind’s speed is 6 mph, what is Kurt’s speed on his bike? #solutionbox[ #math.equation(block: true, alt: "34 mph")[$"34 mph"$] ] Amanda jogs to the park 8 miles using one route and then returns via a 14-mile route. The return trip takes her 1 hour longer than her jog to the park. Find her jogging rate. An experienced window washer can wash all the windows in Mike’s house in 2 hours, while a new trainee can wash all the windows in 7 hours. How long would it take them working together? #solutionbox[ #math.equation(block: true, alt: "1 the fraction 5 over 9 hours")[$1 frac(5, 9) #h(0.2em) "hours"$] ] Josh can split a truckload of logs in 8 hours, but working with his dad they can get it done in 3 hours. How long would it take Josh’s dad working alone to split the logs? The price that Tyler pays for gas varies directly with the number of gallons he buys. If 24 gallons cost him \$59.76, what would 30 gallons cost? #solutionbox[ #math.equation(block: true, alt: "$ 74.70")[$"$" 74.70$] ] The volume of a gas in a container varies inversely with the pressure on the gas. If a container of nitrogen has a volume of 29.5 liters with 2000 psi, what is the volume if the tank has a 14.7 psi rating? Round to the nearest whole number. The cities of Dayton, Columbus, and Cincinnati form a triangle in southern Ohio, as shown on the figure below, that gives the map distances between these cities in inches. #figure(figph[This is an image of a triangle. Clockwise beginning at the top, each vertex is labeled. The top vertex is labeled “Dayton”, the next vertex is labeled “Columbus”, and the next vertex is labeled “Cincinnati”. The distance from Dayton to Columbus is 3.2 inches. The distance from Columbus to Cincinnati is 5.3 inches. The distance from Cincinnati to Dayton is 2.4 inches.], alt: "This is an image of a triangle. Clockwise beginning at the top, each vertex is labeled. The top vertex is labeled “Dayton”, the next vertex is labeled “Columbus”, and the next vertex is labeled “Cincinnati”. The distance from Dayton to Columbus is 3.2 inches. The distance from Columbus to Cincinnati is 5.3 inches. The distance from Cincinnati to Dayton is 2.4 inches.", caption: none) The actual distance from Dayton to Cincinnati is 48 miles. What is the actual distance between Dayton and Columbus? #solutionbox[ 64 miles ]