#set document(title: "9.7 Solve a Formula for a Specific Variable", 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")) == 9.7#h(0.6em)Solve a Formula for a Specific Variable #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Write #math.equation(block: false, alt: "35")[$35$] miles per gallon as a unit rate. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "the fraction 35 miles over 1 gallon")[$frac(35 " " "miles", 1 " " "gallon")$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve #math.equation(block: false, alt: "6 x plus 24 equals 96 .")[$6 x + 24 = 96 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "12")[$12$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the simple interest earned after #math.equation(block: false, alt: "5")[$5$] years on #math.equation(block: false, alt: "$1,000")[$"$1,000"$] at an interest rate of #math.equation(block: false, alt: "4% .")[$"4%" .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "$200")[$\$200$] ] ] === Use the Distance, Rate, and Time Formula One formula you’ll use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant speed. The basic idea is probably already familiar to you. Do you know what distance you travel if you drove at a steady rate of #math.equation(block: false, alt: "60")[$60$] miles per hour for #math.equation(block: false, alt: "2")[$2$] hours? (This might happen if you use your car’s cruise control while driving on the Interstate.) If you said #math.equation(block: false, alt: "120")[$120$] miles, you already know how to use this formula! The math to calculate the distance might look like this: #math.equation(block: true, alt: "distance equals open parenthesis the fraction 60 miles over 1 hour close parenthesis open parenthesis 2 hours close parenthesis; distance equals 120 miles")[$"distance" = ( frac(60 #h(0.2em) "miles", 1 #h(0.2em) "hour") ) ( 2 #h(0.2em) "hours" ) \ "distance" = 120 #h(0.2em) "miles"$]In general, the formula relating distance, rate, and time is #math.equation(block: true, alt: "distance = rate times time")[$"distance" #h(0.2em) "=" #h(0.2em) "rate" · "time"$]#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Distance, Rate and Time] For an object moving in at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula #math.equation(block: true, alt: "d equals r t")[$d = r t$]where #math.equation(block: false, alt: "d equals")[$d =$] distance, #math.equation(block: false, alt: "r equals")[$r =$] rate, and #math.equation(block: false, alt: "t equals")[$t =$] time. ] Notice that the units we used above for the rate were miles per hour, which we can write as a ratio #math.equation(block: false, alt: "the fraction m i l e s over h o u r .")[$frac(m i l e s, h o u r) .$] Then when we multiplied by the time, in hours, the common units ‘hour’ divided out. The answer was in miles. #examplebox("Example 1")[][ Jamal rides his bike at a uniform rate of #math.equation(block: false, alt: "12")[$12$] miles per hour for #math.equation(block: false, alt: "3 the fraction 1 over 2")[$3 frac(1, 2)$] hours. How much distance has he traveled? #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem. #linebreak() You may want to create a mini-chart to summarize the #linebreak() information in the problem.], [#math.equation(block: false, alt: "d equals ?")[$d = ?$] #linebreak() #math.equation(block: false, alt: "r equals 12 mph")[$r = 12 #h(0.2em) "mph"$] #linebreak() #math.equation(block: false, alt: "t equals 3 the fraction 1 over 2 hours")[$t = 3 frac(1, 2) #h(0.2em) "hours"$]]), [Step 2. #strong[Identify] what you are looking for.], [distance traveled], [Step 3. #strong[Name.] Choose a variable to represent it.], [let #emph[d] = distance], [Step 4. #strong[Translate.] #linebreak() Write the appropriate formula for the situation. #linebreak() Substitute in the given information.], [#math.equation(block: false, alt: "d equals r t")[$d = r t$] #linebreak() #linebreak() #math.equation(block: false, alt: "d equals 12 times 3 the fraction 1 over 2")[$d = 12 ⋅ 3 frac(1, 2)$]], [Step 5. #strong[Solve] the equation.], [#math.equation(block: false, alt: "d equals 42 miles")[$d = 42 #h(0.2em) "miles"$]], [Step 6. #strong[Check:] Does 42 miles make sense? #linebreak() #figure(figph[A list shows Jamal rides 12, 24, 36, and 48 miles in 1, 2, 3, and 4 hours respectively. An arrow points to the statement that 42 miles in 3 1/2 hours is reasonable.], alt: "A list shows Jamal rides 12, 24, 36, and 48 miles in 1, 2, 3, and 4 hours respectively. An arrow points to the statement that 42 miles in 3 1/2 hours is reasonable.", caption: none)], [], [Step 7. #strong[Answer] the question with a complete sentence.], [Jamal rode 42 miles.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Lindsay drove for #math.equation(block: false, alt: "5 the fraction 1 over 2")[$5 frac(1, 2)$] hours at #math.equation(block: false, alt: "60")[$60$] miles per hour. How much distance did she travel? #solutionbox[ 330 mi ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Trinh walked for #math.equation(block: false, alt: "2 the fraction 1 over 3")[$2 frac(1, 3)$] hours at #math.equation(block: false, alt: "3")[$3$] miles per hour. How far did she walk? #solutionbox[ 7 mi ] ] #examplebox("Example 2")[][ Rey is planning to drive from his house in San Diego to visit his grandmother in Sacramento, a distance of #math.equation(block: false, alt: "520")[$520$] miles. If he can drive at a steady rate of #math.equation(block: false, alt: "65")[$65$] miles per hour, how many hours will the trip take? #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem. #linebreak() Summarize the information in the problem.], [#math.equation(block: false, alt: "d equals 520 miles")[$d = 520 #h(0.2em) "miles"$] #linebreak() #math.equation(block: false, alt: "r equals 65 mph")[$r = 65 #h(0.2em) "mph"$] #linebreak() #math.equation(block: false, alt: "t equals ?")[$t = ?$]]), [Step 2. #strong[Identify] what you are looking for.], [how many hours (time)], [Step 3. #strong[Name:] #linebreak() Choose a variable to represent it.], [let #emph[t] = time], [Step 4. #strong[Translate.] #linebreak() Write the appropriate formula. #linebreak() Substitute in the given information.], [#math.equation(block: false, alt: "d equals r t")[$d = r t$] #linebreak() #math.equation(block: false, alt: "520 equals 65 t")[$520 = 65 t$]], [Step 5. #strong[Solve] the equation.], [#math.equation(block: false, alt: "t equals 8")[$t = 8$]], [Step 6. #strong[Check:] #linebreak() Substitute the numbers into the formula and make sure #linebreak() the result is a true statement. #linebreak() #math.equation(block: false, alt: "d equals r t")[$d = r t$] #linebreak() #math.equation(block: false, alt: "520 equals ? 65 times 8")[$520 limits(=)^(?) 65 ⋅ 8$] #linebreak() #math.equation(block: false, alt: "520 equals 520> ✓")[$520 = 520> ✓$]], [], [Step 7. #strong[Answer] the question with a complete sentence. #linebreak() We know the units of time will be hours because #linebreak() we divided miles by miles per hour.], [Rey's trip will take 8 hours.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Lee wants to drive from Phoenix to his brother’s apartment in San Francisco, a distance of #math.equation(block: false, alt: "770")[$770$] miles. If he drives at a steady rate of #math.equation(block: false, alt: "70")[$70$] miles per hour, how many hours will the trip take? #solutionbox[ 11 hours ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Yesenia is #math.equation(block: false, alt: "168")[$168$] miles from Chicago. If she needs to be in Chicago in #math.equation(block: false, alt: "3")[$3$] hours, at what rate does she need to drive? #solutionbox[ 56 mph ] ] === Solve a Formula for a Specific Variable In this chapter, you became familiar with some formulas used in geometry. Formulas are also very useful in the sciences and social sciences—fields such as chemistry, physics, biology, psychology, sociology, and criminal justice. Healthcare workers use formulas, too, even for something as routine as dispensing medicine. The widely used spreadsheet program Microsoft Excel#super[TM] relies on formulas to do its calculations. Many teachers use spreadsheets to apply formulas to compute student grades. It is important to be familiar with formulas and be able to manipulate them easily. In and , we used the formula #math.equation(block: false, alt: "d equals r t .")[$d = r t .$] This formula gives the value of #math.equation(block: false, alt: "d")[$d$] when you substitute in the values of #math.equation(block: false, alt: "r")[$r$] and #math.equation(block: false, alt: "t .")[$t .$] But in , we had to find the value of #math.equation(block: false, alt: "t .")[$t .$] We substituted in values of #math.equation(block: false, alt: "d")[$d$] and #math.equation(block: false, alt: "r")[$r$] and then used algebra to solve to #math.equation(block: false, alt: "t .")[$t .$] If you had to do this often, you might wonder why there isn’t a formula that gives the value of #math.equation(block: false, alt: "t")[$t$] when you substitute in the values of #math.equation(block: false, alt: "d")[$d$] and #math.equation(block: false, alt: "r .")[$r .$] We can get a formula like this by solving the formula #math.equation(block: false, alt: "d equals r t")[$d = r t$] for #math.equation(block: false, alt: "t .")[$t .$] #emph[To solve a formula for a specific variable] means to get that variable by itself with a coefficient of #math.equation(block: false, alt: "1")[$1$] on one side of the equation and all the other variables and constants on the other side. We will call this solving an equation for a specific variable #emph[in general.] This process is also called #emph[solving a literal equation]. The result is another formula, made up only of variables. The formula contains letters, or #emph[literals]. Let’s try a few examples, starting with the distance, rate, and time formula we used above. #examplebox("Example 3")[][ Solve the formula #math.equation(block: false, alt: "d equals r t")[$d = r t$] for #math.equation(block: false, alt: "t :")[$t ":"$] + ⓐ when #math.equation(block: false, alt: "d equals 520")[$d = 520$] and #math.equation(block: false, alt: "r equals 65")[$r = 65$] + ⓑ in general. #solutionbox[ We’ll write the solutions side-by-side so you can see that solving a formula in general uses the same steps as when we have numbers to substitute. #figure(table( columns: 3, align: left, inset: 6pt, table.header([], [ⓐ when #emph[d] = 520 and #emph[r] = 65], [ⓑ in general]), [Write the forumla.], [#figure(figph[The mathematical formula d = rt, representing distance equals rate times time, is displayed in bold black text on a plain white background.], alt: "The mathematical formula d = rt, representing distance equals rate times time, is displayed in bold black text on a plain white background.", caption: none)], [#figure(figph[The mathematical formula for distance, rate, and time: d = rt, where 'd' is distance, 'r' is rate (or speed), and 't' is time. This fundamental equation is used in physics and everyday calculations.], alt: "The mathematical formula for distance, rate, and time: d = rt, where 'd' is distance, 'r' is rate (or speed), and 't' is time. This fundamental equation is used in physics and everyday calculations.", caption: none)], [Substitute any given values.], [#figure(figph[A mathematical equation is displayed on a white background, reading '520 = 65t'.], alt: "A mathematical equation is displayed on a white background, reading '520 = 65t'.", caption: none)], [], [Divide to isolate #emph[t].], [#figure(figph[A mathematical equation showing 520 divided by 65 equals 65t divided by 65.], alt: "A mathematical equation showing 520 divided by 65 equals 65t divided by 65.", caption: none)], [#figure(figph[Illustrating the division step to solve for 't' in the equation d = rt, resulting in t = d/r.], alt: "Illustrating the division step to solve for 't' in the equation d = rt, resulting in t = d/r.", caption: none)], [Simplify.], [#figure(figph[The mathematical expressions '8=t' and 't=8' are shown, illustrating that equality is symmetrical.], alt: "The mathematical expressions '8=t' and 't=8' are shown, illustrating that equality is symmetrical.", caption: none)], [#figure(figph[The formula for time (t) derived from distance (d) and rate (r), expressed as d/r = t and t = d/r, representing the same fundamental relationship.], alt: "The formula for time (t) derived from distance (d) and rate (r), expressed as d/r = t and t = d/r, representing the same fundamental relationship.", caption: none)], )) Notice that the solution for ⓐ is the same as that in . We say the formula #math.equation(block: false, alt: "t equals the fraction d over r")[$t = frac(d, r)$] is solved for #math.equation(block: false, alt: "t .")[$t .$] We can use this version of the formula anytime we are given the distance and rate and need to find the time. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the formula #math.equation(block: false, alt: "d equals r t")[$d = r t$] for #math.equation(block: false, alt: "r :")[$r ":"$] + ⓐ when #math.equation(block: false, alt: "d equals 180")[$d = 180$] and #math.equation(block: false, alt: "t equals 4")[$t = 4$] + ⓐ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "r equals 45")[$#h(0.2em) r = 45$] + ⓑ #math.equation(block: false, alt: "r equals the fraction d over t")[$#h(0.2em) r = frac(d, t)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the formula #math.equation(block: false, alt: "d equals r t")[$d = r t$] for #math.equation(block: false, alt: "r :")[$r ":"$] + ⓐ when #math.equation(block: false, alt: "d equals 780")[$d = 780$] and #math.equation(block: false, alt: "t equals 12")[$t = 12$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "r equals 65")[$#h(0.2em) r = 65$] + ⓑ #math.equation(block: false, alt: "r equals the fraction d over t")[$#h(0.2em) r = frac(d, t)$] ] We used the formula #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h")[$A = frac(1, 2) b h$] in Use Properties of Rectangles, Triangles, and Trapezoids to find the area of a triangle when we were given the base and height. In the next example, we will solve this formula for the height. ] #examplebox("Example 4")[][ The formula for area of a triangle is #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h .")[$A = frac(1, 2) b h .$] Solve this formula for #math.equation(block: false, alt: "h :")[$h ":"$] + ⓐ when #math.equation(block: false, alt: "A equals 90")[$A = 90$] and #math.equation(block: false, alt: "b equals 15")[$b = 15$] + ⓑ in general #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, table.header([], [ⓐ when #emph[A] = 90 and #emph[b] = 15], [ⓑ in general]), [Write the forumla.], [#figure(figph[The mathematical formula A = 1/2bh, which represents the area of a triangle where A is the area, b is the base, and h is the height.], alt: "The mathematical formula A = 1/2bh, which represents the area of a triangle where A is the area, b is the base, and h is the height.", caption: none)], [#figure(figph[The formula for the area of a triangle, A = (1/2)bh, where A is the area, b is the base, and h is the height, is displayed on a white background.], alt: "The formula for the area of a triangle, A = (1/2)bh, where A is the area, b is the base, and h is the height, is displayed on a white background.", caption: none)], [Substitute any given values.], [#figure(figph[A mathematical equation reads '90 = 1/2 \* 15 \* h' displayed in black font against a white background.], alt: "A mathematical equation reads '90 = 1/2 * 15 * h' displayed in black font against a white background.", caption: none)], [], [Clear the fractions.], [#figure(figph[A mathematical equation reads 2 multiplied by 90 equals 2 multiplied by 1/2 multiplied by 15 multiplied by h. The number 2 on both sides and the fraction 1/2 are highlighted in red.], alt: "A mathematical equation reads 2 multiplied by 90 equals 2 multiplied by 1/2 multiplied by 15 multiplied by h. The number 2 on both sides and the fraction 1/2 are highlighted in red.", caption: none)], [#figure(figph[A mathematical equation illustrating the formula for the area of a triangle, where both sides of the equation, 2 \* A and 2 \* (1/2) \* b \* h, are multiplied by 2, with the number 2 highlighted in red.], alt: "A mathematical equation illustrating the formula for the area of a triangle, where both sides of the equation, 2 * A and 2 * (1/2) * b * h, are multiplied by 2, with the number 2 highlighted in red.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background: 180 = 15h.], alt: "A mathematical equation is displayed on a white background: 180 = 15h.", caption: none)], [#figure(figph[The image shows the mathematical equation 2A = bh, representing a formula for calculating area in geometry, where A is area, b is base, and h is height.], alt: "The image shows the mathematical equation 2A = bh, representing a formula for calculating area in geometry, where A is area, b is base, and h is height.", caption: none)], [Solve for #emph[h].], [#figure(figph[The equation '12 = h' is displayed on a white background, representing a mathematical statement where the variable h is equal to the number 12.], alt: "The equation '12 = h' is displayed on a white background, representing a mathematical statement where the variable h is equal to the number 12.", caption: none)], [#figure(figph[A mathematical equation shows '2A/b = h' with '2A' over 'b' on the left side, an equals sign in the middle, and 'h' on the right side. The characters are black on a white background.], alt: "A mathematical equation shows '2A/b = h' with '2A' over 'b' on the left side, an equals sign in the middle, and 'h' on the right side. The characters are black on a white background.", caption: none)], )) We can now find the height of a triangle, if we know the area and the base, by using the formula #math.equation(block: true, alt: "h equals the fraction 2 A over b")[$h = frac(2 A, b)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the formula #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h")[$A = frac(1, 2) b h$] to solve for #math.equation(block: false, alt: "h:")[$"h:"$] + ⓐ when #math.equation(block: false, alt: "A equals 170")[$A = 170$] and #math.equation(block: false, alt: "b equals 17")[$b = 17$] + ⓑ in general #solutionbox[ ⓐ #math.equation(block: false, alt: "h equals 20")[$#h(0.2em) h = 20$] ⓑ #math.equation(block: false, alt: "h equals the fraction 2 A over b")[$#h(0.2em) h = frac(2 A, b)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the formula #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h")[$A = frac(1, 2) b h$] to solve for #math.equation(block: false, alt: "b :")[$b ":"$] + ⓐ when #math.equation(block: false, alt: "A equals 62")[$A = 62$] and #math.equation(block: false, alt: "h equals 31")[$h = 31$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "b equals 4")[$#h(0.2em) b = 4$] + ⓑ #math.equation(block: false, alt: "b equals the fraction 2 A over h")[$#h(0.2em) b = frac(2 A, h)$] ] ] In Solve Simple Interest Applications, we used the formula #math.equation(block: false, alt: "I equals P r t")[$I = P r t$] to calculate simple interest, where #math.equation(block: false, alt: "I")[$I$] is interest, #math.equation(block: false, alt: "P")[$P$] is principal, #math.equation(block: false, alt: "r")[$r$] is rate as a decimal, and #math.equation(block: false, alt: "t")[$t$] is time in years. #examplebox("Example 5")[][ Solve the formula #math.equation(block: false, alt: "I equals P r t")[$I = P r t$] to find the principal, #math.equation(block: false, alt: "P :")[$P ":"$] + ⓐ when #math.equation(block: false, alt: "I equals $5,600 , r equals 4% , t equals 7 years")[$I = "$5,600" , #h(0.2em) r = "4%" , #h(0.2em) t = 7 #h(0.2em) "years"$] + ⓑ in general #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, table.header([], [#emph[I] = \$5600, #emph[r] = 4%, #emph[t] = 7 years], [in general]), [Write the forumla.], [#figure(figph[The simple interest formula is displayed, showing I = Prt. This equation represents how to calculate simple interest (I) based on the principal amount (P), the annual interest rate (r), and the time (t) in years.], alt: "The simple interest formula is displayed, showing I = Prt. This equation represents how to calculate simple interest (I) based on the principal amount (P), the annual interest rate (r), and the time (t) in years.", caption: none)], [#figure(figph[The simple interest formula I=Prt is displayed in black text on a white background. 'I' represents interest, 'P' is the principal amount, 'r' is the annual interest rate, and 't' is the time in years.], alt: "The simple interest formula I=Prt is displayed in black text on a white background. 'I' represents interest, 'P' is the principal amount, 'r' is the annual interest rate, and 't' is the time in years.", caption: none)], [Substitute any given values.], [#figure(figph[A mathematical equation showing 5600 equals P multiplied by 0.04 and then by 7. This represents a simple interest calculation where 5600 is the interest, P is the principal, 0.04 is the rate, and 7 is the time.], alt: "A mathematical equation showing 5600 equals P multiplied by 0.04 and then by 7. This represents a simple interest calculation where 5600 is the interest, P is the principal, 0.04 is the rate, and 7 is the time.", caption: none)], [#figure(figph[The simple interest formula I=Prt is displayed in black text on a white background. 'I' represents interest, 'P' is the principal amount, 'r' is the annual interest rate, and 't' is the time in years.], alt: "The simple interest formula I=Prt is displayed in black text on a white background. 'I' represents interest, 'P' is the principal amount, 'r' is the annual interest rate, and 't' is the time in years.", caption: none)], [Multiply #emph[r] ⋅ #emph[t].], [#figure(figph[A mathematical equation is displayed, reading '5600 = P(0.28)', where P likely represents a variable being multiplied by 0.28 to equal 5600.], alt: "A mathematical equation is displayed, reading '5600 = P(0.28)', where P likely represents a variable being multiplied by 0.28 to equal 5600.", caption: none)], [#figure(figph[The image shows the simple interest formula, I = P(rt), where I is interest, P is principal, r is the interest rate, and t is time.], alt: "The image shows the simple interest formula, I = P(rt), where I is interest, P is principal, r is the interest rate, and t is time.", caption: none)], [Divide to isolate #emph[P].], [#figure(figph[A mathematical equation shows '5600 over 0.28 equals P(0.28) over 0.28'. The numerators are 5600 and P(0.28), while the denominators are 0.28 for both sides, with the 0.28 in red.], alt: "A mathematical equation shows '5600 over 0.28 equals P(0.28) over 0.28'. The numerators are 5600 and P(0.28), while the denominators are 0.28 for both sides, with the 0.28 in red.", caption: none)], [#figure(figph[Demonstration of isolating 'P' in the simple interest formula by dividing both sides of I = Prt by 'rt', resulting in I/rt = P(rt)/rt.], alt: "Demonstration of isolating 'P' in the simple interest formula by dividing both sides of I = Prt by 'rt', resulting in I/rt = P(rt)/rt.", caption: none)], [Simplify.], [#figure(figph[An equation displays '20,000 = P' in black bold font against a white background.], alt: "An equation displays '20,000 = P' in black bold font against a white background.", caption: none)], [#figure(figph[A mathematical equation displays I over rt equals P, written as I/(rt) = P, on a white background. The variables are in italic font.], alt: "A mathematical equation displays I over rt equals P, written as I/(rt) = P, on a white background. The variables are in italic font.", caption: none)], [State the answer.], [The principal is \$20,000.], [#figure(figph[A mathematical formula displaying P = I / (rt). This equation relates a principal amount (P) to interest (I), rate (r), and time (t), often seen in financial or physics contexts.], alt: "A mathematical formula displaying P = I / (rt). This equation relates a principal amount (P) to interest (I), rate (r), and time (t), often seen in financial or physics contexts.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the formula #math.equation(block: false, alt: "I equals P r t .")[$I = P r t .$] Find #math.equation(block: false, alt: "t :")[$t ":"$] ⓐ when #math.equation(block: false, alt: "I equals $2,160 , r equals 6% , P equals $12,000;")[$I = "$2,160" , #h(0.2em) r = "6%" , #h(0.2em) P = "$12,000;"$] ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "t equals 3 years")[$#h(0.2em) t = "3 years" #h(0.2em)$] + ⓑ #math.equation(block: false, alt: "t equals the fraction I over P r")[$#h(0.2em) t = frac(I, P #h(0.2em) r)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the formula #math.equation(block: false, alt: "I equals P r t .")[$I = P r t .$] Find #math.equation(block: false, alt: "r :")[$r ":"$] ⓐ when #math.equation(block: false, alt: "I equals $5,400 , P equals $9,000 , t equals 5 years")[$I = "$5,400" , #h(0.2em) P = "$9,000" , #h(0.2em) t = 5 #h(0.2em) "years"$] ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "r equals 0.12 equals 12%")[$#h(0.2em) r = 0.12 = #h(0.2em) "12%" #h(0.2em)$] + ⓑ #math.equation(block: false, alt: "r equals the fraction I over P t")[$#h(0.2em) r = frac(I, P #h(0.2em) t)$] ] ] Later in this class, and in future algebra classes, you’ll encounter equations that relate two variables, usually #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y .")[$y .$] You might be given an equation that is solved for #math.equation(block: false, alt: "y")[$y$] and need to solve it for #math.equation(block: false, alt: "x ,")[$x ,$] or vice versa. In the following example, we’re given an equation with both #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] on the same side and we’ll solve it for #math.equation(block: false, alt: "y .")[$y .$] To do this, we will follow the same steps that we used to solve a formula for a specific variable. #examplebox("Example 6")[][ Solve the formula #math.equation(block: false, alt: "3 x plus 2 y equals 18")[$3 x + 2 y = 18$] for #math.equation(block: false, alt: "y :")[$y ":"$] + ⓐ when #math.equation(block: false, alt: "x equals 4")[$x = 4$] + ⓑ in general #solutionbox[ #figure(table( columns: 3, align: left, inset: 6pt, table.header([], [when #emph[x] = 4], [in general]), [Write the equation.], [#figure(figph[A mathematical equation is displayed, showing '3x + 2y = 18' in black text against a white background.], alt: "A mathematical equation is displayed, showing '3x + 2y = 18' in black text against a white background.", caption: none)], [#figure(figph[A mathematical equation is displayed on a white background, reading '3x + 2y = 18'. The equation is presented in a clear, standard mathematical font.], alt: "A mathematical equation is displayed on a white background, reading '3x + 2y = 18'. The equation is presented in a clear, standard mathematical font.", caption: none)], [Substitute any given values.], [#figure(figph[The image displays the algebraic equation '3(4) + 2y = 18' centered on a white background.], alt: "The image displays the algebraic equation '3(4) + 2y = 18' centered on a white background.", caption: none)], [#figure(figph[A mathematical equation is displayed on a white background, reading '3x + 2y = 18'. The equation is presented in a clear, standard mathematical font.], alt: "A mathematical equation is displayed on a white background, reading '3x + 2y = 18'. The equation is presented in a clear, standard mathematical font.", caption: none)], [Simplify if possible.], [#figure(figph[A mathematical equation is displayed on a white background, which reads '12 + 2y = 18'.], alt: "A mathematical equation is displayed on a white background, which reads '12 + 2y = 18'.", caption: none)], [#figure(figph[A mathematical equation is displayed on a white background, reading '3x + 2y = 18'. The equation is presented in a clear, standard mathematical font.], alt: "A mathematical equation is displayed on a white background, reading '3x + 2y = 18'. The equation is presented in a clear, standard mathematical font.", caption: none)], [Subtract to isolate the #emph[y]-term.], [#figure(figph[An algebraic equation showing a step in solving for 'y', specifically: 12 - 12 + 2y = 18 - 12. The number 12 is highlighted in red where it is being subtracted on both sides.], alt: "An algebraic equation showing a step in solving for 'y', specifically: 12 - 12 + 2y = 18 - 12. The number 12 is highlighted in red where it is being subtracted on both sides.", caption: none)], [#figure(figph[An algebraic equation is shown, with 3x - 3x + 2y on the left side and 18 - 3x on the right. The terms -3x on the left and -3x on the right are highlighted in red, indicating an operation or change.], alt: "An algebraic equation is shown, with 3x - 3x + 2y on the left side and 18 - 3x on the right. The terms -3x on the left and -3x on the right are highlighted in red, indicating an operation or change.", caption: none)], [Simplify.], [#figure(figph[The image displays the algebraic equation '2y = 6' in a clear, standard mathematical notation, presented on a white background.], alt: "The image displays the algebraic equation '2y = 6' in a clear, standard mathematical notation, presented on a white background.", caption: none)], [#figure(figph[A mathematical equation is displayed on a white background: 2y = 18 - 3x.], alt: "A mathematical equation is displayed on a white background: 2y = 18 - 3x.", caption: none)], [Divide.], [#figure(figph[The equation 2y/2 = 6/2 is displayed, with the number 2 in the denominator of both fractions highlighted in red to show division.], alt: "The equation 2y/2 = 6/2 is displayed, with the number 2 in the denominator of both fractions highlighted in red to show division.", caption: none)], [#figure(figph[A mathematical equation showing 2y divided by 2 equals the expression (18 minus 3x) divided by 2, with the number 2 in the denominators highlighted in red.], alt: "A mathematical equation showing 2y divided by 2 equals the expression (18 minus 3x) divided by 2, with the number 2 in the denominators highlighted in red.", caption: none)], [Simplify.], [#figure(figph[The mathematical equation 'y = 3' is displayed in a black serif font against a plain white background, centered in the frame.], alt: "The mathematical equation 'y = 3' is displayed in a black serif font against a plain white background, centered in the frame.", caption: none)], [#figure(figph[A mathematical equation shows y equals a fraction. The numerator is 18 minus 3x, and the denominator is 2.], alt: "A mathematical equation shows y equals a fraction. The numerator is 18 minus 3x, and the denominator is 2.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the formula #math.equation(block: false, alt: "3 x plus 4 y equals 10")[$3 x + 4 y = 10$] for #math.equation(block: false, alt: "y :")[$y ":"$] + ⓐ when #math.equation(block: false, alt: "x equals 2")[$x = 2$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "y equals 1")[$#h(0.2em) y = 1$] + ⓑ #math.equation(block: false, alt: "y equals the fraction 10 minus 3 x over 4")[$#h(0.2em) y = frac(10 − 3 x, 4)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the formula #math.equation(block: false, alt: "5 x plus 2 y equals 18")[$5 x + 2 y = 18$] for #math.equation(block: false, alt: "y :")[$y ":"$] + ⓐ when #math.equation(block: false, alt: "x equals 4")[$x = 4$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "y equals −1")[$#h(0.2em) y = −1$] + ⓑ #math.equation(block: false, alt: "y equals the fraction 18 minus 5 x over 2")[$#h(0.2em) y = frac(18 − 5 x, 2)$] ] ] In the previous examples, we used the numbers in part (a) as a guide to solving in general in part (b). Do you think you’re ready to solve a formula in general without using numbers as a guide? #examplebox("Example 7")[][ Solve the formula #math.equation(block: false, alt: "P equals a plus b plus c")[$P = a + b + c$] for #math.equation(block: false, alt: "a .")[$a .$] #solutionbox[ We will isolate #math.equation(block: false, alt: "a")[$a$] on one side of the equation. #linebreak() #figure(table( columns: 3, align: left, inset: 6pt, table.header([We will isolate #emph[a] on one side of the equation.], [], []), [Write the equation.], [], [#math.equation(block: false, alt: "P equals a plus b plus c")[$P = a + b + c$]], [Subtract #emph[b] and #emph[c] from both sides to isolate #emph[a].], [], [#figure(figph[A mathematical equation is displayed: P - b - c = a + b + c - b - c. The variables 'b' and 'c' are highlighted in red on both sides of the equation, suggesting simplification or cancellation.], alt: "A mathematical equation is displayed: P - b - c = a + b + c - b - c. The variables 'b' and 'c' are highlighted in red on both sides of the equation, suggesting simplification or cancellation.", caption: none)], [Simplify.], [], [#math.equation(block: false, alt: "P minus b minus c equals a")[$P − b − c = a$]], )) So, #math.equation(block: false, alt: "a equals P minus b minus c")[$a = P − b − c$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the formula #math.equation(block: false, alt: "P equals a plus b plus c")[$P = a + b + c$] for #math.equation(block: false, alt: "b .")[$b "."$] #solutionbox[ #emph[b] = #emph[P] − #emph[a] − #emph[c] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the formula #math.equation(block: false, alt: "P equals a plus b plus c")[$P = a + b + c$] for #math.equation(block: false, alt: "c .")[$c "."$] #solutionbox[ #emph[c] = #emph[P] − #emph[a] − #emph[b] ] ] #examplebox("Example 8")[][ Solve the equation #math.equation(block: false, alt: "3 x plus y equals 10")[$3 x + y = 10$] for #math.equation(block: false, alt: "y .")[$y .$] #solutionbox[ We will isolate #math.equation(block: false, alt: "y")[$y$] on one side of the equation. #linebreak() #figure(table( columns: 3, align: left, inset: 6pt, table.header([We will isolate #emph[y] on one side of the equation.], [], []), [Write the equation.], [], [#math.equation(block: false, alt: "3 x plus y equals 10")[$3 x + y = 10$]], [Subtract 3#emph[x] from both sides to isolate #emph[y].], [], [#figure(figph[A mathematical equation showing the subtraction of 3x from both sides: 3x - 3x + y = 10 - 3x. This is a step to isolate the variable y.], alt: "A mathematical equation showing the subtraction of 3x from both sides: 3x - 3x + y = 10 - 3x. This is a step to isolate the variable y.", caption: none)], [Simplify.], [], [#math.equation(block: false, alt: "y equals 10 minus 3 x")[$y = 10 − 3 x$]], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the formula #math.equation(block: false, alt: "7 x plus y equals 11")[$7 x + y = 11$] for #math.equation(block: false, alt: "y .")[$y .$] #solutionbox[ #emph[y] = 11 − 7#emph[x] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the formula #math.equation(block: false, alt: "11 x plus y equals 8")[$11 x + y = 8$] for #math.equation(block: false, alt: "y .")[$y .$] #solutionbox[ #emph[y] = 8 − 11#emph[x] ] ] #examplebox("Example 9")[][ Solve the equation #math.equation(block: false, alt: "6 x plus 5 y equals 13")[$6 x + 5 y = 13$] for #math.equation(block: false, alt: "y .")[$y .$] #solutionbox[ We will isolate #math.equation(block: false, alt: "y")[$y$] on one side of the equation. #linebreak() #figure(table( columns: 2, align: left, inset: 6pt, table.header([We will isolate #emph[y] on one side of the equation.], []), [Write the equation.], [#figure(figph[A linear equation in two variables, 6x + 5y = 13, is displayed in black text on a white background.], alt: "A linear equation in two variables, 6x + 5y = 13, is displayed in black text on a white background.", caption: none)], [Subtract to isolate the term with #emph[y].], [#figure(figph[A math equation: 6x + 5y - 6x = 13 - 6x. The 6x terms being subtracted from both sides are highlighted in red, illustrating a step to simplify the equation and isolate the 5y term.], alt: "A math equation: 6x + 5y - 6x = 13 - 6x. The 6x terms being subtracted from both sides are highlighted in red, illustrating a step to simplify the equation and isolate the 5y term.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is displayed on a white background, reading '5y = 13 - 6x' in black text.], alt: "A mathematical equation is displayed on a white background, reading '5y = 13 - 6x' in black text.", caption: none)], [Divide 5 to make the coefficient 1.], [#figure(figph[A mathematical equation shows 5y/5 = (13 - 6x)/5. The number 5, as the denominator on both sides, is highlighted in red, indicating a division operation.], alt: "A mathematical equation shows 5y/5 = (13 - 6x)/5. The number 5, as the denominator on both sides, is highlighted in red, indicating a division operation.", caption: none)], [Simplify.], [#figure(figph[A mathematical equation is displayed: y = (13 - 6x) / 5.], alt: "A mathematical equation is displayed: y = (13 - 6x) / 5.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the formula #math.equation(block: false, alt: "4 x plus 7 y equals 9")[$4 x + 7 y = 9$] for #math.equation(block: false, alt: "y .")[$y .$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 9 minus 4 x over 7")[$y = frac(9 − 4 x, 7)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve the formula #math.equation(block: false, alt: "5 x plus 8 y equals 1")[$5 x + 8 y = 1$] for #math.equation(block: false, alt: "y .")[$y .$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 1 minus 5 x over 8")[$y = frac(1 − 5 x, 8)$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[ACCESS ADDITIONAL ONLINE RESOURCES] - #link("https://www.openstax.org/l/24distratextime")[#math.equation(block: false, alt: "Distance equals Rate x Time")[$"Distance" #h(0.2em) = #h(0.2em) "Rate" #h(0.2em) "x" #h(0.2em) "Time"$]] - #link("https://www.openstax.org/l/24distratetime")[Distance, Rate, Time] - #link("https://www.openstax.org/l/24simpinterest")[Simple Interest] - #link("https://www.openstax.org/l/24solvespvari")[Solving a Formula for a Specific Variable] - #link("https://www.openstax.org/l/24solvespecivar")[Solving a Formula for a Specific Variable] ] === Key Concepts - #emph[Distance, Rate, and Time] - #math.equation(block: false, alt: "d equals r t")[$d = r t$] === Section Exercises ==== Practice Makes Perfect #strong[Use the Distance, Rate, and Time Formula] In the following exercises, solve. Steve drove for #math.equation(block: false, alt: "8 the fraction 1 over 2")[$8 frac(1, 2)$] hours at #math.equation(block: false, alt: "72")[$72$] miles per hour. How much distance did he travel? #solutionbox[ 612 mi ] Socorro drove for #math.equation(block: false, alt: "4 the fraction 5 over 6")[$4 frac(5, 6)$] hours at #math.equation(block: false, alt: "60")[$60$] miles per hour. How much distance did she travel? Yuki walked for #math.equation(block: false, alt: "1 the fraction 3 over 4")[$1 frac(3, 4)$] hours at #math.equation(block: false, alt: "4")[$4$] miles per hour. How far did she walk? #solutionbox[ 7 mi ] Francie rode her bike for #math.equation(block: false, alt: "2 the fraction 1 over 2")[$2 frac(1, 2)$] hours at #math.equation(block: false, alt: "12")[$12$] miles per hour. How far did she ride? Connor wants to drive from Tucson to the Grand Canyon, a distance of #math.equation(block: false, alt: "338")[$338$] miles. If he drives at a steady rate of #math.equation(block: false, alt: "52")[$52$] miles per hour, how many hours will the trip take? #solutionbox[ 6.5 hours ] Megan is taking the bus from New York City to Montreal. The distance is #math.equation(block: false, alt: "384")[$384$] miles and the bus travels at a steady rate of #math.equation(block: false, alt: "64")[$64$] miles per hour. How long will the bus ride be? Aurelia is driving from Miami to Orlando at a rate of #math.equation(block: false, alt: "65")[$65$] miles per hour. The distance is #math.equation(block: false, alt: "235")[$235$] miles. To the nearest tenth of an hour, how long will the trip take? #solutionbox[ 3.6 hours ] Kareem wants to ride his bike from St. Louis, Missouri to Champaign, Illinois. The distance is #math.equation(block: false, alt: "180")[$180$] miles. If he rides at a steady rate of #math.equation(block: false, alt: "16")[$16$] miles per hour, how many hours will the trip take? Javier is driving to Bangor, Maine, which is #math.equation(block: false, alt: "240")[$240$] miles away from his current location. If he needs to be in Bangor in #math.equation(block: false, alt: "4")[$4$] hours, at what rate does he need to drive? #solutionbox[ 60 mph ] Alejandra is driving to Cincinnati, Ohio, #math.equation(block: false, alt: "450")[$450$] miles away. If she wants to be there in #math.equation(block: false, alt: "6")[$6$] hours, at what rate does she need to drive? Aisha took the train from Spokane to Seattle. The distance is #math.equation(block: false, alt: "280")[$280$] miles, and the trip took #math.equation(block: false, alt: "3.5")[$3.5$] hours. What was the speed of the train? #solutionbox[ 80 mph ] Philip got a ride with a friend from Denver to Las Vegas, a distance of #math.equation(block: false, alt: "750")[$750$] miles. If the trip took #math.equation(block: false, alt: "10")[$10$] hours, how fast was the friend driving? #strong[Solve a Formula for a Specific Variable] In the following exercises, use the formula. #math.equation(block: false, alt: "d equals r t .")[$d = r t .$] Solve for #math.equation(block: false, alt: "t :")[$t ":"$] + ⓐ when #math.equation(block: false, alt: "d equals 350")[$d = 350$] and #math.equation(block: false, alt: "r equals 70")[$r = 70$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "t equals 5")[$#h(0.2em) t = 5$] + ⓑ #math.equation(block: false, alt: "t equals the fraction d over r")[$#h(0.2em) t = frac(d, r)$] ] Solve for #math.equation(block: false, alt: "t :")[$t ":"$] + ⓐ when #math.equation(block: false, alt: "d equals 240")[$d = 240$] and #math.equation(block: false, alt: "r equals 60")[$r = 60$] + ⓑ in general Solve for #math.equation(block: false, alt: "t :")[$t ":"$] + ⓐ when #math.equation(block: false, alt: "d equals 510")[$d = 510$] and #math.equation(block: false, alt: "r equals 60")[$r = 60$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "t equals 8.5")[$#h(0.2em) t = 8.5$] + ⓑ #math.equation(block: false, alt: "t equals the fraction d over r")[$#h(0.2em) t = frac(d, r)$] ] Solve for #math.equation(block: false, alt: "t :")[$t ":"$] + ⓐ when #math.equation(block: false, alt: "d equals 175")[$d = 175$] and #math.equation(block: false, alt: "r equals 50")[$r = 50$] + ⓑ in general Solve for #math.equation(block: false, alt: "r :")[$r ":"$] + ⓐ when #math.equation(block: false, alt: "d equals 204")[$d = 204$] and #math.equation(block: false, alt: "t equals 3")[$t = 3$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "r equals 68")[$#h(0.2em) r = 68$] + ⓑ #math.equation(block: false, alt: "r equals the fraction d over t")[$#h(0.2em) r = frac(d, t)$] ] Solve for #math.equation(block: false, alt: "r :")[$r ":"$] + ⓐ when #math.equation(block: false, alt: "d equals 420")[$d = 420$] and #math.equation(block: false, alt: "t equals 6")[$t = 6$] + ⓑ in general Solve for #math.equation(block: false, alt: "r :")[$r ":"$] + ⓐ when #math.equation(block: false, alt: "d equals 160")[$d = 160$] and #math.equation(block: false, alt: "t equals 2.5")[$t = 2.5$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "r equals 64")[$#h(0.2em) r = 64$] + ⓑ #math.equation(block: false, alt: "r equals the fraction d over t")[$#h(0.2em) r = frac(d, t)$] ] Solve for #math.equation(block: false, alt: "r :")[$r ":"$] + ⓐ when #math.equation(block: false, alt: "d equals 180")[$d = 180$] and #math.equation(block: false, alt: "t equals 4.5")[$t = 4.5$] + ⓑ in general. In the following exercises, use the formula #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h .")[$A = frac(1, 2) b h .$] Solve for #math.equation(block: false, alt: "b :")[$b ":"$] + ⓐ when #math.equation(block: false, alt: "A equals 126")[$A = 126$] and #math.equation(block: false, alt: "h equals 18")[$h = 18$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "b equals 14")[$#h(0.2em) b = 14$] + ⓑ #math.equation(block: false, alt: "b equals the fraction 2 A over h")[$#h(0.2em) b = frac(2 A, h)$] ] Solve for #math.equation(block: false, alt: "h :")[$h ":"$] + ⓐ when #math.equation(block: false, alt: "A equals 176")[$A = 176$] and #math.equation(block: false, alt: "b equals 22")[$b = 22$] + ⓑ in general Solve for #math.equation(block: false, alt: "h :")[$h ":"$] + ⓐ when #math.equation(block: false, alt: "A equals 375")[$A = 375$] and #math.equation(block: false, alt: "b equals 25")[$b = 25$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "h equals 30")[$#h(0.2em) h = 30$] + ⓑ #math.equation(block: false, alt: "h equals the fraction 2 A over b")[$#h(0.2em) h = frac(2 A, b)$] ] Solve for #math.equation(block: false, alt: "b :")[$b ":"$] + ⓐ when #math.equation(block: false, alt: "A equals 65")[$A = 65$] and #math.equation(block: false, alt: "h equals 13")[$h = 13$] + ⓑ in general In the following exercises, use the formula #math.equation(block: false, alt: "I equals P r t .")[$I = P r t .$] Solve for the principal, #math.equation(block: false, alt: "P")[$P$] for: + ⓐ #math.equation(block: false, alt: "I equals $5,480")[$#h(0.2em) I = "$5,480"$], #math.equation(block: false, alt: "r equals 4%")[$r = "4%"$], #math.equation(block: false, alt: "t equals 7 years")[$t = 7 #h(0.2em) "years"$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "P equals $19,571.43")[$#h(0.2em) P = "$19,571.43"$] + ⓑ #math.equation(block: false, alt: "P equals the fraction I over r t")[$#h(0.2em) P = frac(I, r t)$] ] Solve for the principal, #math.equation(block: false, alt: "P")[$P$] for: + ⓐ #math.equation(block: false, alt: "I equals $3,950")[$#h(0.2em) I = "$3,950"$], #math.equation(block: false, alt: "r equals 6%")[$r = "6%"$], #math.equation(block: false, alt: "t equals 5 years")[$t = 5 #h(0.2em) "years"$] + ⓑ in general Solve for the time, #math.equation(block: false, alt: "t")[$t$] for: + ⓐ #math.equation(block: false, alt: "I equals $2,376")[$#h(0.2em) I = "$2,376"$], #math.equation(block: false, alt: "P equals $9,000")[$P = "$9,000"$], #math.equation(block: false, alt: "r equals 4.4 %")[$r = 4.4 %$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "t equals 6 years")[$#h(0.2em) t = 6 #h(0.2em) "years"$] + ⓑ #math.equation(block: false, alt: "t equals the fraction I over P r")[$#h(0.2em) t = frac(I, P r)$] ] Solve for the time, #emph[#math.equation(block: false, alt: "t")[$t$]] for: + ⓐ #math.equation(block: false, alt: "I equals $624")[$#h(0.2em) I = "$624"$], #math.equation(block: false, alt: "P equals $6,000")[$P = "$6,000"$], #math.equation(block: false, alt: "r equals 5.2 %")[$r = 5.2 %$] + ⓑ in general In the following exercises, solve. Solve the formula #math.equation(block: false, alt: "2 x plus 3 y equals 12")[$2 x + 3 y = 12$] for #math.equation(block: false, alt: "y :")[$y ":"$] + ⓐ when #math.equation(block: false, alt: "x equals 3")[$x = 3$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "y equals 2")[$#h(0.2em) y = 2$] + ⓑ #math.equation(block: false, alt: "y equals the fraction 12 minus 2 x over 3")[$#h(0.2em) y = frac(12 − 2 x, 3)$] ] Solve the formula #math.equation(block: false, alt: "5 x plus 2 y equals 10")[$5 x + 2 y = 10$] for #math.equation(block: false, alt: "y :")[$y ":"$] + ⓐ when #math.equation(block: false, alt: "x equals 4")[$x = 4$] + ⓑ in general Solve the formula #math.equation(block: false, alt: "3 x plus y equals 7")[$3 x + y = 7$] for #math.equation(block: false, alt: "y :")[$y ":"$] + ⓐ when #math.equation(block: false, alt: "x equals −2")[$x = −2$] + ⓑ in general #solutionbox[ + ⓐ #emph[y] = 13 + ⓑ #emph[y] = 7 − 3#emph[x] ] Solve the formula #math.equation(block: false, alt: "4 x plus y equals 5")[$4 x + y = 5$] for #math.equation(block: false, alt: "y :")[$y ":"$] + ⓐ when #math.equation(block: false, alt: "x equals −3")[$x = −3$] + ⓑ in general Solve #math.equation(block: false, alt: "a plus b equals 90")[$a + b = 90$] for #math.equation(block: false, alt: "b .")[$b .$] #solutionbox[ + #emph[b] = 90 − #emph[a] ] Solve #math.equation(block: false, alt: "a plus b equals 90")[$a + b = 90$] for #math.equation(block: false, alt: "a .")[$a .$] Solve #math.equation(block: false, alt: "180 equals a plus b plus c")[$180 = a + b + c$] for #math.equation(block: false, alt: "a .")[$a .$] #solutionbox[ #emph[a] = 180 − #emph[b] − #emph[c] ] Solve #math.equation(block: false, alt: "180 equals a plus b plus c")[$180 = a + b + c$] for #math.equation(block: false, alt: "c .")[$c .$] Solve the formula #math.equation(block: false, alt: "8 x plus y equals 15")[$8 x + y = 15$] for #math.equation(block: false, alt: "y .")[$y .$] #solutionbox[ #emph[y] = 15 − 8#emph[x] ] Solve the formula #math.equation(block: false, alt: "9 x plus y equals 13")[$9 x + y = 13$] for #math.equation(block: false, alt: "y .")[$y .$] Solve the formula #math.equation(block: false, alt: "minus 4 x plus y equals −6")[$− 4 x + y = −6$] for #math.equation(block: false, alt: "y .")[$y .$] #solutionbox[ #emph[y] = −6 + 4#emph[x] ] Solve the formula #math.equation(block: false, alt: "minus 5 x plus y equals −1")[$− 5 x + y = −1$] for #math.equation(block: false, alt: "y .")[$y .$] Solve the formula #math.equation(block: false, alt: "4 x plus 3 y equals 7")[$4 x + 3 y = 7$] for #math.equation(block: false, alt: "y .")[$y .$] #solutionbox[ #math.equation(block: true, alt: "y equals the fraction 7 minus 4 x over 3")[$y = frac(7 − 4 x, 3)$] ] Solve the formula #math.equation(block: false, alt: "3 x plus 2 y equals 11")[$3 x + 2 y = 11$] for #math.equation(block: false, alt: "y .")[$y .$] Solve the formula #math.equation(block: false, alt: "x minus y equals −4")[$x − y = −4$] for #math.equation(block: false, alt: "y .")[$y .$] #solutionbox[ #emph[y] = 4 + #emph[x] ] Solve the formula #math.equation(block: false, alt: "x minus y equals −3")[$x − y = −3$] for #math.equation(block: false, alt: "y .")[$y .$] Solve the formula #math.equation(block: false, alt: "P equals 2 L plus 2 W")[$P = 2 L + 2 W$] for #math.equation(block: false, alt: "L .")[$L .$] #solutionbox[ #math.equation(block: true, alt: "L equals the fraction P minus 2 W over 2")[$L = frac(P − 2 W, 2)$] ] Solve the formula #math.equation(block: false, alt: "P equals 2 L plus 2 W")[$P = 2 L + 2 W$] for #math.equation(block: false, alt: "W .")[$W .$] Solve the formula #math.equation(block: false, alt: "C equals π d")[$C = π d$] for #math.equation(block: false, alt: "d .")[$d .$] #solutionbox[ #math.equation(block: true, alt: "d equals the fraction C over π")[$d = frac(C, π)$] ] Solve the formula #math.equation(block: false, alt: "C equals π d")[$C = π d$] for #math.equation(block: false, alt: "π .")[$π .$] Solve the formula #math.equation(block: false, alt: "V equals L W H")[$V = L W H$] for #math.equation(block: false, alt: "L .")[$L .$] #solutionbox[ #math.equation(block: true, alt: "L equals the fraction V over W H")[$L = frac(V, W H)$] ] Solve the formula #math.equation(block: false, alt: "V equals L W H")[$V = L W H$] for #math.equation(block: false, alt: "H .")[$H .$] ==== Everyday Math #strong[Converting temperature] While on a tour in Greece, Tatyana saw that the temperature was #math.equation(block: false, alt: "40°")[$"40°"$] Celsius. Solve for #math.equation(block: false, alt: "F")[$F$] in the formula #math.equation(block: false, alt: "C equals the fraction 5 over 9 open parenthesis F minus 32 close parenthesis")[$C = frac(5, 9) ( F − 32 )$] to find the temperature in Fahrenheit. #solutionbox[ 104° F ] #strong[Converting temperature] Yon was visiting the United States and he saw that the temperature in Seattle was #math.equation(block: false, alt: "50°")[$"50°"$] Fahrenheit. Solve for #math.equation(block: false, alt: "C")[$C$] in the formula #math.equation(block: false, alt: "F equals the fraction 9 over 5 C plus 32")[$F = frac(9, 5) C + 32$] to find the temperature in Celsius. ==== Writing Exercises Solve the equation #math.equation(block: false, alt: "2 x plus 3 y equals 6")[$2 x + 3 y = 6$] for #math.equation(block: false, alt: "y :")[$y ":"$] + ⓐ when #math.equation(block: false, alt: "x equals −3")[$x = −3$] + ⓑ in general + ⓒ Which solution is easier for you? Explain why. #solutionbox[ Answers will vary ] Solve the equation #math.equation(block: false, alt: "5 x minus 2 y equals 10")[$5 x − 2 y = 10$] for #math.equation(block: false, alt: "x :")[$x ":"$] + ⓐ when #math.equation(block: false, alt: "y equals 10")[$y = 10$] + ⓑ in general + ⓒ Which solution is easier for you? Explain why. ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[A self-assessment rubric for math, where students evaluate their proficiency in using the distance, rate, and time formula, and solving equations for variables, using options like 'Confidently' or 'No-I don't get it!'], alt: "A self-assessment rubric for math, where students evaluate their proficiency in using the distance, rate, and time formula, and solving equations for variables, using options like 'Confidently' or 'No-I don't get it!'", caption: none) ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not? === Chapter Review Exercises ==== Use a Problem Solving Strategy #strong[Approach Word Problems with a Positive Attitude] In the following exercises, solve. How has your attitude towards solving word problems changed as a result of working through this chapter? Explain. #solutionbox[ Answers will vary. ] Did the Problem Solving Strategy help you solve word problems in this chapter? Explain. #strong[Use a Problem Solving Strategy for Word Problems] In the following exercises, solve using the problem-solving strategy for word problems. Remember to write a complete sentence to answer each question. Three-fourths of the people at a concert are children. If there are #math.equation(block: false, alt: "87")[$87$] children, what is the total number of people at the concert? #solutionbox[ There are 116 people at the concert. ] There are #math.equation(block: false, alt: "9")[$9$] saxophone players in the band. The number of saxophone players is one less than twice the number of tuba players. Find the number of tuba players. Reza was very sick and lost #math.equation(block: false, alt: "15%")[$"15%"$] of his original weight. He lost #math.equation(block: false, alt: "27")[$27$] pounds. What was his original weight? #solutionbox[ His original weight was 180 pounds. ] Dolores bought a crib on sale for #math.equation(block: false, alt: "$350 .")[$"$350" .$] The sale price was #math.equation(block: false, alt: "40%")[$"40%"$] of the original price. What was the original price of the crib? #strong[Solve Number Problems] In the following exercises, solve each number word problem. The sum of a number and three is forty-one. Find the number. #solutionbox[ 38 ] Twice the difference of a number and ten is fifty-four. Find the number. One number is nine less than another. Their sum is twenty-seven. Find the numbers. #solutionbox[ 18, 9 ] The sum of two consecutive integers is #math.equation(block: false, alt: "minus 135 .")[$− 135 .$] Find the numbers. ==== Solve Money Applications #strong[Solve Coin Word Problems] In the following exercises, solve each coin word problem. Francie has #math.equation(block: false, alt: "$4.35")[$"$4.35"$] in dimes and quarters. The number of dimes is #math.equation(block: false, alt: "5")[$5$] more than the number of quarters. How many of each coin does she have? #solutionbox[ 16 dimes, 11 quarters ] Scott has #math.equation(block: false, alt: "$0.39")[$"$0.39"$] in pennies and nickels. The number of pennies is #math.equation(block: false, alt: "8")[$8$] times the number of nickels. How many of each coin does he have? Paulette has #math.equation(block: false, alt: "$140")[$"$140"$] in #math.equation(block: false, alt: "$5")[$"$5"$] and #math.equation(block: false, alt: "$10")[$"$10"$] bills. The number of #math.equation(block: false, alt: "$10")[$"$10"$] bills is one less than twice the number of #math.equation(block: false, alt: "$5")[$"$5"$] bills. How many of each does she have? #solutionbox[ 6 of \$5 bills, 11 of \$10 bills ] Lenny has #math.equation(block: false, alt: "$3.69")[$"$3.69"$] in pennies, dimes, and quarters. The number of pennies is #math.equation(block: false, alt: "3")[$3$] more than the number of dimes. The number of quarters is twice the number of dimes. How many of each coin does he have? #strong[Solve Ticket and Stamp Word Problems] In the following exercises, solve each ticket or stamp word problem. A church luncheon made #math.equation(block: false, alt: "$842 .")[$"$842" .$] Adult tickets cost #math.equation(block: false, alt: "$10")[$"$10"$] each and children’s tickets cost #math.equation(block: false, alt: "$6")[$"$6"$] each. The number of children was #math.equation(block: false, alt: "12")[$12$] more than twice the number of adults. How many of each ticket were sold? #solutionbox[ 35 adults, 82 children ] Tickets for a basketball game cost #math.equation(block: false, alt: "$2")[$"$2"$] for students and #math.equation(block: false, alt: "$5")[$"$5"$] for adults. The number of students was #math.equation(block: false, alt: "3")[$3$] less than #math.equation(block: false, alt: "10")[$10$] times the number of adults. The total amount of money from ticket sales was #math.equation(block: false, alt: "$619 .")[$"$619" .$] How many of each ticket were sold? Ana spent #math.equation(block: false, alt: "$4.06")[$"$4.06"$] buying stamps. The number of #math.equation(block: false, alt: "$0.41")[$"$0.41"$] stamps she bought was #math.equation(block: false, alt: "5")[$5$] more than the number of #math.equation(block: false, alt: "$0.26")[$"$0.26"$] stamps. How many of each did she buy? #solutionbox[ 3 of 26 -cent stamps, 8 of 41 -cent stamps ] Yumi spent #math.equation(block: false, alt: "$34.15")[$"$34.15"$] buying stamps. The number of #math.equation(block: false, alt: "$0.56")[$"$0.56"$] stamps she bought was #math.equation(block: false, alt: "10")[$10$] less than #math.equation(block: false, alt: "4")[$4$] times the number of #math.equation(block: false, alt: "$0.41")[$"$0.41"$] stamps. How many of each did she buy? ==== Use Properties of Angles, Triangles, and the Pythagorean Theorem #strong[Use Properties of Angles] In the following exercises, solve using properties of angles. What is the supplement of a #math.equation(block: false, alt: "48°")[$"48°"$] angle? #solutionbox[ 132° ] What is the complement of a #math.equation(block: false, alt: "61°")[$"61°"$] angle? Two angles are complementary. The smaller angle is #math.equation(block: false, alt: "24°")[$"24°"$] less than the larger angle. Find the measures of both angles. #solutionbox[ 33°, 57° ] Two angles are supplementary. The larger angle is #math.equation(block: false, alt: "45°")[$"45°"$] more than the smaller angle. Find the measures of both angles. #strong[Use Properties of Triangles] In the following exercises, solve using properties of triangles. The measures of two angles of a triangle are #math.equation(block: false, alt: "22")[$22$] and #math.equation(block: false, alt: "85")[$85$] degrees. Find the measure of the third angle. #solutionbox[ 73° ] One angle of a right triangle measures #math.equation(block: false, alt: "41.5")[$41.5$] degrees. What is the measure of the other small angle? One angle of a triangle is #math.equation(block: false, alt: "30°")[$"30°"$] more than the smallest angle. The largest angle is the sum of the other angles. Find the measures of all three angles. #solutionbox[ 30°, 60°, 90° ] One angle of a triangle is twice the measure of the smallest angle. The third angle is #math.equation(block: false, alt: "60°")[$"60°"$] more than the measure of the smallest angle. Find the measures of all three angles. In the following exercises, #math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] is similar to #math.equation(block: false, alt: "Δ X Y Z .")[$"Δ" X Y Z .$] Find the length of the indicated side. #figure(figph[Two triangles are shown. Triangle ABC is on the left. The side across from A is labeled 21, across from B is b, and across from C is 11.2. Triangle XYZ is on the right. The side across from X is labeled x, across from Y is 10, and across from Z is 8.], alt: "Two triangles are shown. Triangle ABC is on the left. The side across from A is labeled 21, across from B is b, and across from C is 11.2. Triangle XYZ is on the right. The side across from X is labeled x, across from Y is 10, and across from Z is 8.", caption: none) side #math.equation(block: false, alt: "x")[$x$] #solutionbox[ 15 ] side #math.equation(block: false, alt: "b")[$b$] #strong[Use the Pythagorean Theorem] In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary. #figure(figph[A right triangle is shown. The base is labeled 10, the height is labeled 24.], alt: "A right triangle is shown. The base is labeled 10, the height is labeled 24.", caption: none) #solutionbox[ 26 ] #figure(figph[A right triangle is shown. The base is labeled 6, the height is labeled 8.], alt: "A right triangle is shown. The base is labeled 6, the height is labeled 8.", caption: none) #figure(figph[A right triangle is shown. The height is labeled 15, the hypotenuse is labeled 17.], alt: "A right triangle is shown. The height is labeled 15, the hypotenuse is labeled 17.", caption: none) #solutionbox[ 8 ] #figure(figph[A right triangle is shown. The height is labeled 15, the hypotenuse is labeled 25.], alt: "A right triangle is shown. The height is labeled 15, the hypotenuse is labeled 25.", caption: none) #figure(figph[A right triangle is shown. The height is labeled 7, the base is labeled 4.], alt: "A right triangle is shown. The height is labeled 7, the base is labeled 4.", caption: none) #solutionbox[ 8.1 ] #figure(figph[A right triangle is shown. The height is labeled 11, the base is labeled 10.], alt: "A right triangle is shown. The height is labeled 11, the base is labeled 10.", caption: none) In the following exercises, solve. Approximate to the nearest tenth, if necessary. Sergio needs to attach a wire to hold the antenna to the roof of his house, as shown in the figure. The antenna is #math.equation(block: false, alt: "8")[$8$] feet tall and Sergio has #math.equation(block: false, alt: "10")[$10$] feet of wire. How far from the base of the antenna can he attach the wire? #figure(figph[An image of a house is shown. A 10-foot wire is going from the roof of the house to the ground. The wire hits the house at a height of 8 feet.], alt: "An image of a house is shown. A 10-foot wire is going from the roof of the house to the ground. The wire hits the house at a height of 8 feet.", caption: none) #solutionbox[ 6 feet ] Seong is building shelving in his garage. The shelves are #math.equation(block: false, alt: "36")[$36$] inches wide and #math.equation(block: false, alt: "15")[$15$] inches tall. He wants to put a diagonal brace across the back to stabilize the shelves, as shown. How long should the brace be? #figure(figph[A rectangular shelf is shown, with a diagonal drawn in from the lower left corner to the upper right corner. The side is labeled 15 inches, the top is labeled 36 inches.], alt: "A rectangular shelf is shown, with a diagonal drawn in from the lower left corner to the upper right corner. The side is labeled 15 inches, the top is labeled 36 inches.", caption: none) ==== Use Properties of Rectangles, Triangles, and Trapezoids #strong[Understand Linear, Square, Cubic Measure] In the following exercises, would you measure each item using linear, square, or cubic measure? amount of sand in a sandbag #solutionbox[ cubic ] height of a tree size of a patio #solutionbox[ square ] length of a highway In the following exercises, find + ⓐ the perimeter + ⓑ the area of each figure #figure(figph[Three squares are shown, in a sideways L shape.], alt: "Three squares are shown, in a sideways L shape.", caption: none) #solutionbox[ + ⓐ 8 units + ⓑ 3 sq. units ] #figure(figph[Five squares are shown, in a T-shape. There are three squares across the top and three squares down.], alt: "Five squares are shown, in a T-shape. There are three squares across the top and three squares down.", caption: none) #strong[Use Properties of Rectangles] In the following exercises, find the ⓐ perimeter ⓑ area of each rectangle The length of a rectangle is #math.equation(block: false, alt: "42")[$42$] meters and the width is #math.equation(block: false, alt: "28")[$28$] meters. #solutionbox[ + ⓐ 140 m + ⓑ 1176 sq. m ] The length of a rectangle is #math.equation(block: false, alt: "36")[$36$] feet and the width is #math.equation(block: false, alt: "19")[$19$] feet. A sidewalk in front of Kathy’s house is in the shape of a rectangle #math.equation(block: false, alt: "4")[$4$] feet wide by #math.equation(block: false, alt: "45")[$45$] feet long. #solutionbox[ + ⓐ 98 ft. + ⓑ 180 sq. ft. ] A rectangular room is #math.equation(block: false, alt: "16")[$16$] feet wide by #math.equation(block: false, alt: "12")[$12$] feet long. In the following exercises, solve. Find the length of a rectangle with perimeter of #math.equation(block: false, alt: "220")[$220$] centimeters and width of #math.equation(block: false, alt: "85")[$85$] centimeters. #solutionbox[ 25 cm ] Find the width of a rectangle with perimeter #math.equation(block: false, alt: "39")[$39$] and length #math.equation(block: false, alt: "11 .")[$11 .$] The area of a rectangle is #math.equation(block: false, alt: "2356")[$2356$] square meters. The length is #math.equation(block: false, alt: "38")[$38$] meters. What is the width? #solutionbox[ 62 m ] The width of a rectangle is #math.equation(block: false, alt: "45")[$45$] centimeters. The area is #math.equation(block: false, alt: "2700")[$2700$] square centimeters. What is the length? The length of a rectangle is #math.equation(block: false, alt: "12")[$12$] centimeters more than the width. The perimeter is #math.equation(block: false, alt: "74")[$74$] centimeters. Find the length and the width. #solutionbox[ 24.5 cm., 12.5 cm. ] The width of a rectangle is #math.equation(block: false, alt: "3")[$3$] more than twice the length. The perimeter is #math.equation(block: false, alt: "96")[$96$] inches. Find the length and the width. #strong[Use Properties of Triangles] In the following exercises, solve using the properties of triangles. Find the area of a triangle with base #math.equation(block: false, alt: "18")[$18$] inches and height #math.equation(block: false, alt: "15")[$15$] inches. #solutionbox[ 135 sq. in. ] Find the area of a triangle with base #math.equation(block: false, alt: "33")[$33$] centimeters and height #math.equation(block: false, alt: "21")[$21$] centimeters. A triangular road sign has base #math.equation(block: false, alt: "30")[$30$] inches and height #math.equation(block: false, alt: "40")[$40$] inches. What is its area? #solutionbox[ 600 sq. in. ] If a triangular courtyard has sides #math.equation(block: false, alt: "9")[$9$] feet and #math.equation(block: false, alt: "12")[$12$] feet and the perimeter is #math.equation(block: false, alt: "32")[$32$] feet, how long is the third side? A tile in the shape of an isosceles triangle has a base of #math.equation(block: false, alt: "6")[$6$] inches. If the perimeter is #math.equation(block: false, alt: "20")[$20$] inches, find the length of each of the other sides. #solutionbox[ 7 in., 7 in. ] Find the length of each side of an equilateral triangle with perimeter of #math.equation(block: false, alt: "81")[$81$] yards. The perimeter of a triangle is #math.equation(block: false, alt: "59")[$59$] feet. One side of the triangle is #math.equation(block: false, alt: "3")[$3$] feet longer than the shortest side. The third side is #math.equation(block: false, alt: "5")[$5$] feet longer than the shortest side. Find the length of each side. #solutionbox[ 17 ft., 20 ft., 22 ft. ] One side of a triangle is three times the smallest side. The third side is #math.equation(block: false, alt: "9")[$9$] feet more than the shortest side. The perimeter is #math.equation(block: false, alt: "39")[$39$] feet. Find the lengths of all three sides. #strong[Use Properties of Trapezoids] In the following exercises, solve using the properties of trapezoids. The height of a trapezoid is #math.equation(block: false, alt: "8")[$8$] feet and the bases are #math.equation(block: false, alt: "11")[$11$] and #math.equation(block: false, alt: "14")[$14$] feet. What is the area? #solutionbox[ 100 sq. ft. ] The height of a trapezoid is #math.equation(block: false, alt: "5")[$5$] yards and the bases are #math.equation(block: false, alt: "7")[$7$] and #math.equation(block: false, alt: "10")[$10$] yards. What is the area? Find the area of the trapezoid with height #math.equation(block: false, alt: "25")[$25$] meters and bases #math.equation(block: false, alt: "32.5")[$32.5$] and #math.equation(block: false, alt: "21.5")[$21.5$] meters. #solutionbox[ 675 sq. m ] A flag is shaped like a trapezoid with height #math.equation(block: false, alt: "62")[$62$] centimeters and the bases are #math.equation(block: false, alt: "91.5")[$91.5$] and #math.equation(block: false, alt: "78.1")[$78.1$] centimeters. What is the area of the flag? ==== Solve Geometry Applications: Circles and Irregular Figures #strong[Use Properties of Circles] In the following exercises, solve using the properties of circles. Round answers to the nearest hundredth. A circular mosaic has radius #math.equation(block: false, alt: "3")[$3$] meters. Find the + ⓐ circumference + ⓑ area of the mosaic #solutionbox[ + ⓐ 18.84 m + ⓑ 28.26 sq. m ] A circular fountain has radius #math.equation(block: false, alt: "8")[$8$] feet. Find the + ⓐ circumference + ⓑ area of the fountain Find the diameter of a circle with circumference #math.equation(block: false, alt: "150.72")[$150.72$] inches. #solutionbox[ 48 in. ] Find the radius of a circle with circumference #math.equation(block: false, alt: "345.4")[$345.4$] centimeters #strong[Find the Area of Irregular Figures] In the following exercises, find the area of each shaded region. #figure(figph[A geometric shape is shown, formed by two rectangles. The top is labeled 8. The width of the top rectangle is labeled 3. The right side of the figure is labeled 5. The width of the bottom rectangle is labeled 3.], alt: "A geometric shape is shown, formed by two rectangles. The top is labeled 8. The width of the top rectangle is labeled 3. The right side of the figure is labeled 5. The width of the bottom rectangle is labeled 3.", caption: none) #solutionbox[ 30 sq. units ] #figure(figph[A geometric shape is shown. It is a U-shape. The base is labeled 5, the height 6. The horizontal and vertical lines at the top are labeled 2.], alt: "A geometric shape is shown. It is a U-shape. The base is labeled 5, the height 6. The horizontal and vertical lines at the top are labeled 2.", caption: none) #figure(figph[A geometric shape is shown. It is formed by two triangles. The shared base of the two triangles is labeled 20. The height of each triangle is labeled 15.], alt: "A geometric shape is shown. It is formed by two triangles. The shared base of the two triangles is labeled 20. The height of each triangle is labeled 15.", caption: none) #solutionbox[ 300 sq. units ] #figure(figph[A geometric shape is shown. It is a trapezoid with a triangle attached to the top on the right side. The height of the trapezoid is labeled 8, the bottom base is labeled 12, and the top is labeled 9. The height of the triangle is labeled 8.], alt: "A geometric shape is shown. It is a trapezoid with a triangle attached to the top on the right side. The height of the trapezoid is labeled 8, the bottom base is labeled 12, and the top is labeled 9. The height of the triangle is labeled 8.", caption: none) #figure(figph[A geometric shape is shown. It is a rectangle with a semi-circle attached to the top. The base of the rectangle, also the diameter of the semi-circle, is labeled 10. The height of the rectangle is labeled 16.], alt: "A geometric shape is shown. It is a rectangle with a semi-circle attached to the top. The base of the rectangle, also the diameter of the semi-circle, is labeled 10. The height of the rectangle is labeled 16.", caption: none) #solutionbox[ 199.25 sq. units ] #figure(figph[A geometric shape is shown. It is a triangle with a semicircle attached. The base of the triangle, also the diameter of the semi-circle, is labeled 5. The height of the triangle is also labeled 5.], alt: "A geometric shape is shown. It is a triangle with a semicircle attached. The base of the triangle, also the diameter of the semi-circle, is labeled 5. The height of the triangle is also labeled 5.", caption: none) ==== Solve Geometry Applications: Volume and Surface Area #strong[Find Volume and Surface Area of Rectangular Solids] In the following exercises, find the + ⓐ volume + ⓑ surface area of the rectangular solid a rectangular solid with length #math.equation(block: false, alt: "14")[$14$] centimeters, width #math.equation(block: false, alt: "4.5")[$4.5$] centimeters, and height #math.equation(block: false, alt: "10")[$10$] centimeters #solutionbox[ + ⓐ 630 cu. cm + ⓑ 496 sq. cm ] a cube with sides that are #math.equation(block: false, alt: "3")[$3$] feet long a cube of tofu with sides #math.equation(block: false, alt: "2.5")[$2.5$] inches #solutionbox[ + ⓐ 15.625 cu. in. + ⓑ 37.5 sq. in. ] a rectangular carton with length #math.equation(block: false, alt: "32")[$32$] inches, width #math.equation(block: false, alt: "18")[$18$] inches, and height #math.equation(block: false, alt: "10")[$10$] inches #strong[Find Volume and Surface Area of Spheres] In the following exercises, find the + ⓐ volume + ⓑ surface area of the sphere. a sphere with radius #math.equation(block: false, alt: "4")[$4$] yards #solutionbox[ + ⓐ 267.95 cu. yd. + ⓑ 200.96 sq. yd. ] a sphere with radius #math.equation(block: false, alt: "12")[$12$] meters a baseball with radius #math.equation(block: false, alt: "1.45")[$1.45$] inches #solutionbox[ + ⓐ 12.76 cu. in. + ⓑ 26.41 sq. in. ] a soccer ball with radius #math.equation(block: false, alt: "22")[$22$] centimeters #strong[Find Volume and Surface Area of Cylinders] In the following exercises, find the + ⓐ volume + ⓑ surface area of the cylinder a cylinder with radius #math.equation(block: false, alt: "2")[$2$] yards and height #math.equation(block: false, alt: "6")[$6$] yards #solutionbox[ + ⓐ 75.36 cu. yd. + ⓑ 100.48 sq. yd. ] a cylinder with diameter #math.equation(block: false, alt: "18")[$18$] inches and height #math.equation(block: false, alt: "40")[$40$] inches a juice can with diameter #math.equation(block: false, alt: "8")[$8$] centimeters and height #math.equation(block: false, alt: "15")[$15$] centimeters #solutionbox[ + ⓐ 753.6 cu. cm + ⓑ 477.28 sq. cm ] a cylindrical pylon with diameter #math.equation(block: false, alt: "0.8")[$0.8$] feet and height #math.equation(block: false, alt: "2.5")[$2.5$] feet #strong[Find Volume of Cones] In the following exercises, find the volume of the cone. a cone with height #math.equation(block: false, alt: "5")[$5$] meters and radius #math.equation(block: false, alt: "1")[$1$] meter #solutionbox[ 5.233 cu. m ] a cone with height #math.equation(block: false, alt: "24")[$24$] feet and radius #math.equation(block: false, alt: "8")[$8$] feet a cone-shaped water cup with diameter #math.equation(block: false, alt: "2.6")[$2.6$] inches and height #math.equation(block: false, alt: "2.6")[$2.6$] inches #solutionbox[ 4.599 cu. in. ] a cone-shaped pile of gravel with diameter #math.equation(block: false, alt: "6")[$6$] yards and height #math.equation(block: false, alt: "5")[$5$] yards ==== Solve a Formula for a Specific Variable #strong[Use the Distance, Rate, and Time Formula] In the following exercises, solve using the formula for distance, rate, and time. A plane flew #math.equation(block: false, alt: "4")[$4$] hours at #math.equation(block: false, alt: "380")[$380$] miles per hour. What distance was covered? #solutionbox[ 1520 miles ] Gus rode his bike for #math.equation(block: false, alt: "1 the fraction 1 over 2")[$1 frac(1, 2)$] hours at #math.equation(block: false, alt: "8")[$8$] miles per hour. How far did he ride? Jack is driving from Bangor to Portland at a rate of #math.equation(block: false, alt: "68")[$68$] miles per hour. The distance is #math.equation(block: false, alt: "107")[$107$] miles. To the nearest tenth of an hour, how long will the trip take? #solutionbox[ 1.6 hours ] Jasmine took the bus from Pittsburgh to Philadelphia. The distance is #math.equation(block: false, alt: "305")[$305$] miles and the trip took #math.equation(block: false, alt: "5")[$5$] hours. What was the speed of the bus? #strong[Solve a Formula for a Specific Variable] In the following exercises, use the formula #math.equation(block: false, alt: "d equals r t .")[$d = r t .$] Solve for #math.equation(block: false, alt: "t :")[$t ":"$] + ⓐ when #math.equation(block: false, alt: "d equals 403")[$d = 403$] and #math.equation(block: false, alt: "r equals 65")[$r = 65$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "t equals 6.2")[$#h(0.2em) t = 6.2$] + ⓑ #math.equation(block: false, alt: "t equals the fraction d over r")[$#h(0.2em) t = frac(d, r)$] ] Solve for #math.equation(block: false, alt: "r :")[$r ":"$] + ⓐ when #math.equation(block: false, alt: "d equals 750")[$d = 750$] and #math.equation(block: false, alt: "t equals 15")[$t = 15$] + ⓑ in general In the following exercises, use the formula #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h .")[$A = frac(1, 2) b h .$] Solve for #math.equation(block: false, alt: "b :")[$b ":"$] + ⓐ when #math.equation(block: false, alt: "A equals 416")[$A = 416$] and #math.equation(block: false, alt: "h equals 32")[$h = 32$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "b equals 26")[$#h(0.2em) b = 26$] + ⓑ #math.equation(block: false, alt: "b equals the fraction 2 A over h")[$#h(0.2em) b = frac(2 A, h)$] ] Solve for #math.equation(block: false, alt: "h :")[$h ":"$] + ⓐ when #math.equation(block: false, alt: "A equals 48")[$A = 48$] and #math.equation(block: false, alt: "b equals 8")[$b = 8$] + ⓑ in general In the following exercises, use the formula #math.equation(block: false, alt: "I equals P r t .")[$I = P r t .$] Solve for the principal, #math.equation(block: false, alt: "P ,")[$P ,$] for: + ⓐ #math.equation(block: false, alt: "I equals $720")[$#h(0.2em) I = "$720"$], #math.equation(block: false, alt: "r equals 4%")[$r = "4%"$], #math.equation(block: false, alt: "t equals 3 years")[$t = 3 #h(0.2em) "years"$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "P equals $6000")[$#h(0.2em) P = "$6000"$] + ⓑ #math.equation(block: false, alt: "P equals the fraction I over open parenthesis r times t close parenthesis")[$#h(0.2em) P = frac(I, ( r ⋅ t ))$] ] Solve for the time, #math.equation(block: false, alt: "t")[$t$] for: + ⓐ #math.equation(block: false, alt: "I equals $3630")[$#h(0.2em) I = "$3630"$], #math.equation(block: false, alt: "P equals $11,000")[$P = "$11,000"$], #math.equation(block: false, alt: "r equals 5.5 %")[$r = 5.5 %$] + ⓑ in general In the following exercises, solve. Solve the formula #math.equation(block: false, alt: "6 x plus 5 y equals 20")[$6 x + 5 y = 20$] for #math.equation(block: false, alt: "y :")[$y ":"$] + ⓐ when #math.equation(block: false, alt: "x equals 0")[$x = 0$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "y equals 4")[$#h(0.2em) y = 4$] + ⓑ #math.equation(block: false, alt: "y equals the fraction 20 minus 6 x over 5")[$#h(0.2em) y = frac(20 − 6 x, 5)$] ] Solve the formula #math.equation(block: false, alt: "2 x plus y equals 15")[$2 x + y = 15$] for #math.equation(block: false, alt: "y :")[$y ":"$] + ⓐ when #math.equation(block: false, alt: "x equals −5")[$x = −5$] + ⓑ in general Solve #math.equation(block: false, alt: "a plus b equals 90")[$a + b = 90$] for #math.equation(block: false, alt: "a .")[$a .$] #solutionbox[ #emph[a] = 90 − #emph[b] ] Solve #math.equation(block: false, alt: "180 equals a plus b plus c")[$180 = a + b + c$] for #math.equation(block: false, alt: "a .")[$a .$] Solve the formula #math.equation(block: false, alt: "4 x plus y equals 17")[$4 x + y = 17$] for #math.equation(block: false, alt: "y .")[$y .$] #solutionbox[ #emph[y] = 17 − 4#emph[x] ] Solve the formula #math.equation(block: false, alt: "minus 3 x plus y equals −6")[$− 3 x + y = −6$] for #math.equation(block: false, alt: "y .")[$y .$] Solve the formula #math.equation(block: false, alt: "P equals 2 L plus 2 W")[$P = 2 L + 2 W$] for #math.equation(block: false, alt: "W .")[$W .$] #solutionbox[ #math.equation(block: true, alt: "W equals the fraction P minus 2 L over 2")[$W = frac(P − 2 L, 2)$] ] Solve the formula #math.equation(block: false, alt: "V equals L W H")[$V = L W H$] for #math.equation(block: false, alt: "H .")[$H .$] Describe how you have used two topics from this chapter in your life outside of math class during the past month. === Chapter Practice Test Four-fifths of the people on a hike are children. If there are #math.equation(block: false, alt: "12")[$12$] children, what is the total number of people on the hike? The sum of #math.equation(block: false, alt: "13")[$13$] and twice a number is #math.equation(block: false, alt: "minus 19 .")[$− 19 .$] Find the number. #solutionbox[ −16 ] One number is #math.equation(block: false, alt: "3")[$3$] less than another number. Their sum is #math.equation(block: false, alt: "65 .")[$65 .$] Find the numbers. Bonita has #math.equation(block: false, alt: "$2.95")[$"$2.95"$] in dimes and quarters in her pocket. If she has #math.equation(block: false, alt: "5")[$5$] more dimes than quarters, how many of each coin does she have? #solutionbox[ 7 quarters, 12 dimes ] At a concert, #math.equation(block: false, alt: "$1600")[$"$1600"$] in tickets were sold. Adult tickets were #math.equation(block: false, alt: "$9")[$"$9"$] each and children’s tickets were #math.equation(block: false, alt: "$4")[$"$4"$] each. If the number of adult tickets was #math.equation(block: false, alt: "30")[$30$] fewer than twice the number of children’s tickets, how many of each kind were sold? Find the complement of a #math.equation(block: false, alt: "52°")[$"52°"$] angle. #solutionbox[ 38° ] The measure of one angle of a triangle is twice the measure of the smallest angle. The measure of the third angle is #math.equation(block: false, alt: "14")[$14$] more than the measure of the smallest angle. Find the measures of all three angles. The perimeter of an equilateral triangle is #math.equation(block: false, alt: "145")[$145$] feet. Find the length of each side. #solutionbox[ 48.3 ] #math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] is similar to #math.equation(block: false, alt: "Δ X Y Z .")[$"Δ" X Y Z .$] Find the length of side #math.equation(block: false, alt: "c .")[$c .$] #figure(figph[Two triangles are shown. Triangle XYZ is on the left. The side across from X is labeled 5, the side across from Y is labeled 10, the side across from Z is labeled 7. Triangle ABC is on the right. The side across from A is labeled 6, the side across from B is labeled 12, and the side across from C is labeled c.], alt: "Two triangles are shown. Triangle XYZ is on the left. The side across from X is labeled 5, the side across from Y is labeled 10, the side across from Z is labeled 7. Triangle ABC is on the right. The side across from A is labeled 6, the side across from B is labeled 12, and the side across from C is labeled c.", caption: none) Find the length of the missing side. Round to the nearest tenth, if necessary. #figure(figph[A right triangle is shown. The height is labeled 24 and the hypotenuse is labeled 26.], alt: "A right triangle is shown. The height is labeled 24 and the hypotenuse is labeled 26.", caption: none) #solutionbox[ 10 ] Find the length of the missing side. Round to the nearest tenth, if necessary. #figure(figph[A right triangle is shown. The base is labeled 6 and the height is labeled 9.], alt: "A right triangle is shown. The base is labeled 6 and the height is labeled 9.", caption: none) A baseball diamond is shaped like a square with sides #math.equation(block: false, alt: "90")[$90$] feet long. How far is it from home plate to second base, as shown? #figure(figph[A baseball diamond is shown. It is in the shape of a sideways square. The bottom corner is labeled Home and there is a dotted line to the top corner, labeled 2nd base. The right corner is labeled 1st base and the left corner is labeled 3rd base.], alt: "A baseball diamond is shown. It is in the shape of a sideways square. The bottom corner is labeled Home and there is a dotted line to the top corner, labeled 2nd base. The right corner is labeled 1st base and the left corner is labeled 3rd base.", caption: none) #solutionbox[ 127.3 ft ] The length of a rectangle is #math.equation(block: false, alt: "2")[$2$] feet more than five times the width. The perimeter is #math.equation(block: false, alt: "40")[$40$] feet. Find the dimensions of the rectangle. A triangular poster has base #math.equation(block: false, alt: "80")[$80$] centimeters and height #math.equation(block: false, alt: "55")[$55$] centimeters. Find the area of the poster. #solutionbox[ 2200 square centimeters ] A trapezoid has height #math.equation(block: false, alt: "14")[$14$] inches and bases #math.equation(block: false, alt: "20")[$20$] inches and #math.equation(block: false, alt: "23")[$23$] inches. Find the area of the trapezoid. A circular pool has diameter #math.equation(block: false, alt: "90")[$90$] inches. What is its circumference? Round to the nearest #emph[tenth.] #solutionbox[ 282.6 inches ] Find the area of the shaded region. Round to the nearest tenth. #figure(figph[A geometric shape is shown. It is a rectangle with a semi-circle attached on the left and a triangle attached on the right. The height of the rectangle, also the height of the triangle and the diameter of the semi-circle, is labeled 4. The base of the figure is labeled 10. The top of the rectangle is labeled 7.], alt: "A geometric shape is shown. It is a rectangle with a semi-circle attached on the left and a triangle attached on the right. The height of the rectangle, also the height of the triangle and the diameter of the semi-circle, is labeled 4. The base of the figure is labeled 10. The top of the rectangle is labeled 7.", caption: none) Find the volume of a rectangular room with width #math.equation(block: false, alt: "12")[$12$] feet, length #math.equation(block: false, alt: "15")[$15$] feet, and height #math.equation(block: false, alt: "8")[$8$] feet. #solutionbox[ 1440 ] A coffee can is shaped like a cylinder with height #math.equation(block: false, alt: "7")[$7$] inches and radius #math.equation(block: false, alt: "5")[$5$] inches. Find (a) the surface area and (b) the volume of the can. Round to the nearest tenth. A traffic cone has height #math.equation(block: false, alt: "75")[$75$] centimeters. The radius of the base is #math.equation(block: false, alt: "20")[$20$] centimeters. Find the volume of the cone. Round to the nearest tenth. #solutionbox[ 31,400 cubic inches ] Leon drove from his house in Cincinnati to his sister’s house in Cleveland. He drove at a uniform rate of #math.equation(block: false, alt: "63")[$63$] miles per hour and the trip took #math.equation(block: false, alt: "4")[$4$] hours. What was the distance? The Catalina Express takes #math.equation(block: false, alt: "1 the fraction 1 over 2")[$1 frac(1, 2)$] hours to travel from Long Beach to Catalina Island, a distance of #math.equation(block: false, alt: "22")[$22$] miles. To the nearest tenth, what is the speed of the boat? #solutionbox[ 14.7 miles per hour ] Use the formula #math.equation(block: false, alt: "I equals P r t")[$I = P r t$] to solve for the principal, #math.equation(block: false, alt: "P ,")[$P ,$] for: + ⓐ #math.equation(block: false, alt: "I equals $1380 , r equals 5% , t equals 3")[$#h(0.2em) I = "$1380" , #h(0.2em) r = "5%" , #h(0.2em) t = 3$] years + ⓑ in general Solve the formula #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h")[$A = frac(1, 2) b h$] for #math.equation(block: false, alt: "h :")[$h ":"$] + ⓐ when #math.equation(block: false, alt: "A equals 1716")[$A = 1716$] and #math.equation(block: false, alt: "b equals 66")[$b = 66$] + ⓑ in general #solutionbox[ + ⓐ #math.equation(block: false, alt: "height equals 52")[$#h(0.2em) "height" = 52$] + ⓑ #math.equation(block: false, alt: "h equals the fraction 2 A over b")[$#h(0.2em) h = frac(2 A, b)$] ] Solve #math.equation(block: false, alt: "x plus 5 y equals 14")[$x + 5 y = 14$] for #math.equation(block: false, alt: "y .")[$y .$]