Solve a Formula for a Specific Variable

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 $60$ miles per hour for $2$ hours? (This might happen if you use your car’s cruise control while driving on the Interstate.) If you said $120$ miles, you already know how to use this formula!

The math to calculate the distance might look like this:

\begin{matrix} distance = (\frac{60 miles}{1 hour}) (2 hours) \\ distance = 120 miles \end{matrix}

In general, the formula relating distance, rate, and time is

distance = rate \cdot time

Notice that the units we used above for the rate were miles per hour, which we can write as a ratio $\frac{m i l e s}{h o u r} .$ Then when we multiplied by the time, in hours, the common units ‘hour’ divided out. The answer was in miles.

Jamal rides his bike at a uniform rate of $12$ miles per hour for $3 \frac{1}{2}$ hours. How much distance has he traveled?

Solution

Step 1. Read the problem. You may want to create a mini-chart to summarize the information in the problem.	$d = ?$ $r = 12 mph$ $t = 3 \frac{1}{2} hours$
Step 2. Identify what you are looking for.	distance traveled
Step 3. Name. Choose a variable to represent it.	let d = distance
Step 4. Translate. Write the appropriate formula for the situation. Substitute in the given information.	$d = r t$ $d = 12 \cdot 3 \frac{1}{2}$
Step 5. Solve the equation.	$d = 42 miles$
Step 6. Check: Does 42 miles make sense?
Step 7. Answer the question with a complete sentence.	Jamal rode 42 miles.

Rey is planning to drive from his house in San Diego to visit his grandmother in Sacramento, a distance of $520$ miles. If he can drive at a steady rate of $65$ miles per hour, how many hours will the trip take?

Solution

A seven-step guide to solving a distance, rate, and time problem.
Step 1. Read the problem. Summarize the information in the problem.	$d = 520 miles$ $r = 65 mph$ $t = ?$
Step 2. Identify what you are looking for.	how many hours (time)
Step 3. Name: Choose a variable to represent it.	let t = time
Step 4. Translate. Write the appropriate formula. Substitute in the given information.	$d = r t$ $520 = 65 t$
Step 5. Solve the equation.	$t = 8$
Step 6. Check: Substitute the numbers into the formula and make sure the result is a true statement. $d = r t$ $520 \overset{?}{=} 65 \cdot 8$ $520 = 520> ✓$
Step 7. Answer the question with a complete sentence. We know the units of time will be hours because we divided miles by miles per hour.	Rey's trip will take 8 hours.

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^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 Example 1 and Example 2, we used the formula $d = r t .$ This formula gives the value of $d$ when you substitute in the values of $r$ and $t .$ But in Example 2, we had to find the value of $t .$ We substituted in values of $d$ and $r$ and then used algebra to solve to $t .$ If you had to do this often, you might wonder why there isn’t a formula that gives the value of $t$ when you substitute in the values of $d$ and $r .$ We can get a formula like this by solving the formula $d = r t$ for $t .$

To solve a formula for a specific variable means to get that variable by itself with a coefficient of $1$ 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 in general. This process is also called solving a literal equation. The result is another formula, made up only of variables. The formula contains letters, or literals.

Let’s try a few examples, starting with the distance, rate, and time formula we used above.

Solve the formula

d = r t

for

t :

ⓐ when $d = 520$ and $r = 65$
ⓑ in general.

Solution

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.

Steps to solve d=rt for time (t), showing both a specific numerical example (d=520, r=65) and the general algebraic solution.
	ⓐ when d = 520 and r = 65	ⓑ in general
Write the forumla.
Substitute any given values.
Divide to isolate t.
Simplify.

Notice that the solution for ⓐ is the same as that in Example 2. We say the formula $t = \frac{d}{r}$ is solved for $t .$ We can use this version of the formula anytime we are given the distance and rate and need to find the time.

The formula for area of a triangle is

A = \frac{1}{2} b h .

Solve this formula for

h :

ⓐ when $A = 90$ and $b = 15$
ⓑ in general

Solution

Comparison of step-by-step solutions for a mathematical formula, showing both a specific case (A=90, b=15) and a general derivation.
	ⓐ when A = 90 and b = 15	ⓑ in general
Write the forumla.
Substitute any given values.
Clear the fractions.	$A mathematical equation reads 2 multiplied by 90 equals 2 multiplied by 1/2 multiplied by 15 multiplied by h. The number 2 on both sides and the fraction 1/2 are highlighted in red.$
Simplify.
Solve for h.

We can now find the height of a triangle, if we know the area and the base, by using the formula

h = \frac{2 A}{b}

In Solve Simple Interest Applications, we used the formula $I = P r t$ to calculate simple interest, where $I$ is interest, $P$ is principal, $r$ is rate as a decimal, and $t$ is time in years.

Solve the formula

I = P r t

to find the principal,

P :

ⓐ when $I = $5,600, r = 4%, t = 7 years$
ⓑ in general

Solution

This table illustrates the step-by-step process of calculating the principal (P) from simple interest, providing both a specific numerical example and the general formula derivation.
	I = $5600, r = 4%, t = 7 years	in general
Write the forumla.
Substitute any given values.
Multiply r ⋅ t.
Divide to isolate P.
Simplify.
State the answer.	The principal is $20,000.

Later in this class, and in future algebra classes, you’ll encounter equations that relate two variables, usually $x$ and $y .$ You might be given an equation that is solved for $y$ and need to solve it for $x,$ or vice versa. In the following example, we’re given an equation with both $x$ and $y$ on the same side and we’ll solve it for $y .$ To do this, we will follow the same steps that we used to solve a formula for a specific variable.

Solve the formula

3 x + 2 y = 18

for

y :

ⓐ when $x = 4$
ⓑ in general

Solution

	when x = 4	in general
Write the equation.
Substitute any given values.
Simplify if possible.
Subtract to isolate the y-term.
Simplify.
Divide.	$The equation 2y/2 = 6/2 is displayed, with the number 2 in the denominator of both fractions highlighted in red to show division.$
Simplify.		$A mathematical equation shows y equals a fraction. The numerator is 18 minus 3x, and the denominator is 2.$

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?

Solve the formula $P = a + b + c$ for $a .$

Solution

We will isolate

a

on one side of the equation.

We will isolate a on one side of the equation.
Write the equation.		$P = a + b + c$
Subtract b and c from both sides to isolate a.
Simplify.		$P - b - c = a$

So, $a = P - b - c$

Solve the equation $3 x + y = 10$ for $y .$

Solution

We will isolate

y

on one side of the equation.

We will isolate y on one side of the equation.
Write the equation.		$3 x + y = 10$
Subtract 3x from both sides to isolate y.
Simplify.		$y = 10 - 3 x$

Solve the equation $6 x + 5 y = 13$ for $y .$

Solution

We will isolate

y

on one side of the equation.

We will isolate y on one side of the equation.
Write the equation.
Subtract to isolate the term with y.
Simplify.
Divide 5 to make the coefficient 1.
Simplify.

Key Concepts

Distance, Rate, and Time
- $d = r t$

Section Exercises

Practice Makes Perfect

Use the Distance, Rate, and Time Formula

In the following exercises, solve.

Steve drove for $8 \frac{1}{2}$ hours at $72$ miles per hour. How much distance did he travel?

Solution

612 mi

Socorro drove for $4 \frac{5}{6}$ hours at $60$ miles per hour. How much distance did she travel?

Yuki walked for $1 \frac{3}{4}$ hours at $4$ miles per hour. How far did she walk?

Solution

7 mi

Francie rode her bike for $2 \frac{1}{2}$ hours at $12$ miles per hour. How far did she ride?

Connor wants to drive from Tucson to the Grand Canyon, a distance of $338$ miles. If he drives at a steady rate of $52$ miles per hour, how many hours will the trip take?

Solution

6.5 hours

Megan is taking the bus from New York City to Montreal. The distance is $384$ miles and the bus travels at a steady rate of $64$ miles per hour. How long will the bus ride be?

Aurelia is driving from Miami to Orlando at a rate of $65$ miles per hour. The distance is $235$ miles. To the nearest tenth of an hour, how long will the trip take?

Solution

3.6 hours

Kareem wants to ride his bike from St. Louis, Missouri to Champaign, Illinois. The distance is $180$ miles. If he rides at a steady rate of $16$ miles per hour, how many hours will the trip take?

Javier is driving to Bangor, Maine, which is $240$ miles away from his current location. If he needs to be in Bangor in $4$ hours, at what rate does he need to drive?

Solution

60 mph

Alejandra is driving to Cincinnati, Ohio, $450$ miles away. If she wants to be there in $6$ hours, at what rate does she need to drive?

Aisha took the train from Spokane to Seattle. The distance is $280$ miles, and the trip took $3.5$ hours. What was the speed of the train?

Solution

80 mph

Philip got a ride with a friend from Denver to Las Vegas, a distance of $750$ miles. If the trip took $10$ hours, how fast was the friend driving?

Solve a Formula for a Specific Variable

In the following exercises, use the formula. $d = r t .$

Solve for

t :

ⓐ when $d = 350$ and $r = 70$
ⓑ in general

Solution

ⓐ $t = 5$
ⓑ $t = \frac{d}{r}$

Solve for

t :

ⓐ when $d = 240$ and $r = 60$
ⓑ in general

Solve for

t :

ⓐ when $d = 510$ and $r = 60$
ⓑ in general

Solution

ⓐ $t = 8.5$
ⓑ $t = \frac{d}{r}$

Solve for

t :

ⓐ when $d = 175$ and $r = 50$
ⓑ in general

Solve for

r :

ⓐ when $d = 204$ and $t = 3$
ⓑ in general

Solution

ⓐ $r = 68$
ⓑ $r = \frac{d}{t}$

Solve for

r :

ⓐ when $d = 420$ and $t = 6$
ⓑ in general

Solve for

r :

ⓐ when $d = 160$ and $t = 2.5$
ⓑ in general

Solution

ⓐ $r = 64$
ⓑ $r = \frac{d}{t}$

Solve for

r :

ⓐ when $d = 180$ and $t = 4.5$
ⓑ in general.

In the following exercises, use the formula $A = \frac{1}{2} b h .$

Solve for

b :

ⓐ when $A = 126$ and $h = 18$
ⓑ in general

Solution

ⓐ $b = 14$
ⓑ $b = \frac{2 A}{h}$

Solve for

h :

ⓐ when $A = 176$ and $b = 22$
ⓑ in general

Solve for

h :

ⓐ when $A = 375$ and $b = 25$
ⓑ in general

Solution

ⓐ $h = 30$
ⓑ $h = \frac{2 A}{b}$

Solve for

b :

ⓐ when $A = 65$ and $h = 13$
ⓑ in general

In the following exercises, use the formula $I = P r t .$

Solve for the principal,

P

for:

ⓐ $I = $5,480$ , $r = 4%$ , $t = 7 years$
ⓑ in general

Solution

ⓐ $P = $19,571.43$
ⓑ $P = \frac{I}{r t}$

Solve for the principal,

P

for:

ⓐ $I = $3,950$ , $r = 6%$ , $t = 5 years$
ⓑ in general

Solve for the time,

t

for:

ⓐ $I = $2,376$ , $P = $9,000$ , $r = 4.4 %$
ⓑ in general

Solution

ⓐ $t = 6 years$
ⓑ $t = \frac{I}{P r}$

Solve for the time, $t$ for:

ⓐ $I = $624$ , $P = $6,000$ , $r = 5.2 %$
ⓑ in general

In the following exercises, solve.

Solve the formula

2 x + 3 y = 12

for

y :

ⓐ when $x = 3$
ⓑ in general

Solution

ⓐ $y = 2$
ⓑ $y = \frac{12 - 2 x}{3}$

Solve the formula

5 x + 2 y = 10

for

y :

ⓐ when $x = 4$
ⓑ in general

Solve the formula

3 x + y = 7

for

y :

ⓐ when $x = −2$
ⓑ in general

Solution

ⓐ y = 13
ⓑ y = 7 − 3x

Solve the formula

4 x + y = 5

for

y :

ⓐ when $x = −3$
ⓑ in general

Solve $a + b = 90$ for $b .$

Solution

b = 90 − a

Solve $a + b = 90$ for $a .$

Solve $180 = a + b + c$ for $a .$

Solution

a = 180 − b − c

Solve $180 = a + b + c$ for $c .$

Solve the formula $8 x + y = 15$ for $y .$

Solution

y = 15 − 8x

Solve the formula $9 x + y = 13$ for $y .$

Solve the formula $- 4 x + y = −6$ for $y .$

Solution

y = −6 + 4x

Solve the formula $- 5 x + y = −1$ for $y .$

Solve the formula $4 x + 3 y = 7$ for $y .$

Solution

$y = \frac{7 - 4 x}{3}$

Solve the formula $3 x + 2 y = 11$ for $y .$

Solve the formula $x - y = −4$ for $y .$

Solution

y = 4 + x

Solve the formula $x - y = −3$ for $y .$

Solve the formula $P = 2 L + 2 W$ for $L .$

Solution

$L = \frac{P - 2 W}{2}$

Solve the formula $P = 2 L + 2 W$ for $W .$

Solve the formula $C = π d$ for $d .$

Solution

$d = \frac{C}{π}$

Solve the formula $C = π d$ for $π .$

Solve the formula $V = L W H$ for $L .$

Solution

$L = \frac{V}{W H}$

Solve the formula $V = L W H$ for $H .$

Everyday Math

Converting temperature While on a tour in Greece, Tatyana saw that the temperature was $40°$ Celsius. Solve for $F$ in the formula $C = \frac{5}{9} (F - 32)$ to find the temperature in Fahrenheit.

Solution

104° F

Converting temperature Yon was visiting the United States and he saw that the temperature in Seattle was $50°$ Fahrenheit. Solve for $C$ in the formula $F = \frac{9}{5} C + 32$ to find the temperature in Celsius.

Writing Exercises

Solve the equation

2 x + 3 y = 6

for

y :

ⓐ when $x = −3$
ⓑ in general
ⓒ Which solution is easier for you? Explain why.

Solution

Answers will vary

Solve the equation

5 x - 2 y = 10

for

x :

ⓐ when $y = 10$
ⓑ 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.

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!'

ⓑ 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

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.

Solution

Answers will vary.

Did the Problem Solving Strategy help you solve word problems in this chapter? Explain.

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 $87$ children, what is the total number of people at the concert?

Solution

There are 116 people at the concert.

There are $9$ 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 $15%$ of his original weight. He lost $27$ pounds. What was his original weight?

Solution

His original weight was 180 pounds.

Dolores bought a crib on sale for $$350 .$ The sale price was $40%$ of the original price. What was the original price of the crib?

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.

Solution

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.

Solution

18, 9

The sum of two consecutive integers is $- 135 .$ Find the numbers.

Solve Money Applications

Solve Coin Word Problems

In the following exercises, solve each coin word problem.

Francie has $$4.35$ in dimes and quarters. The number of dimes is $5$ more than the number of quarters. How many of each coin does she have?

Solution

16 dimes, 11 quarters

Scott has $$0.39$ in pennies and nickels. The number of pennies is $8$ times the number of nickels. How many of each coin does he have?

Paulette has $$140$ in $$5$ and $$10$ bills. The number of $$10$ bills is one less than twice the number of $$5$ bills. How many of each does she have?

Solution

6 of $5 bills, 11 of $10 bills

Lenny has $$3.69$ in pennies, dimes, and quarters. The number of pennies is $3$ more than the number of dimes. The number of quarters is twice the number of dimes. How many of each coin does he have?

Solve Ticket and Stamp Word Problems

In the following exercises, solve each ticket or stamp word problem.

A church luncheon made $$842 .$ Adult tickets cost $$10$ each and children’s tickets cost $$6$ each. The number of children was $12$ more than twice the number of adults. How many of each ticket were sold?

Solution

35 adults, 82 children

Tickets for a basketball game cost $$2$ for students and $$5$ for adults. The number of students was $3$ less than $10$ times the number of adults. The total amount of money from ticket sales was $$619 .$ How many of each ticket were sold?

Ana spent $$4.06$ buying stamps. The number of $$0.41$ stamps she bought was $5$ more than the number of $$0.26$ stamps. How many of each did she buy?

Solution

3 of 26 -cent stamps, 8 of 41 -cent stamps

Yumi spent $$34.15$ buying stamps. The number of $$0.56$ stamps she bought was $10$ less than $4$ times the number of $$0.41$ stamps. How many of each did she buy?

Use Properties of Angles, Triangles, and the Pythagorean Theorem

Use Properties of Angles

In the following exercises, solve using properties of angles.

What is the supplement of a $48°$ angle?

Solution

132°

What is the complement of a $61°$ angle?

Two angles are complementary. The smaller angle is $24°$ less than the larger angle. Find the measures of both angles.

Solution

33°, 57°

Two angles are supplementary. The larger angle is $45°$ more than the smaller angle. Find the measures of both angles.

Use Properties of Triangles

In the following exercises, solve using properties of triangles.

The measures of two angles of a triangle are $22$ and $85$ degrees. Find the measure of the third angle.

Solution

73°

One angle of a right triangle measures $41.5$ degrees. What is the measure of the other small angle?

One angle of a triangle is $30°$ more than the smallest angle. The largest angle is the sum of the other angles. Find the measures of all three angles.

Solution

30°, 60°, 90°

One angle of a triangle is twice the measure of the smallest angle. The third angle is $60°$ more than the measure of the smallest angle. Find the measures of all three angles.

In the following exercises, $Δ A B C$ is similar to $Δ X Y Z .$ Find the length of the indicated side.

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.

side $x$

Solution

side $b$

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.

A right triangle is shown. The base is labeled 10, the height is labeled 24.

Solution

A right triangle is shown. The base is labeled 6, the height is labeled 8.

A right triangle is shown. The height is labeled 15, the hypotenuse is labeled 17.

Solution

A right triangle is shown. The height is labeled 15, the hypotenuse is labeled 25.

A right triangle is shown. The height is labeled 7, the base is labeled 4.

Solution

8.1

A right triangle is shown. The height is labeled 11, the base is labeled 10.

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 $8$ feet tall and Sergio has $10$ feet of wire. How far from the base of the antenna can he attach the wire?

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.

Solution

6 feet

Seong is building shelving in his garage. The shelves are $36$ inches wide and $15$ inches tall. He wants to put a diagonal brace across the back to stabilize the shelves, as shown. How long should the brace be?

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.

Use Properties of Rectangles, Triangles, and Trapezoids

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

Solution

cubic

height of a tree

size of a patio

Solution

square

length of a highway

In the following exercises, find

ⓐ the perimeter
ⓑ the area of each figure

Three squares are shown, in a sideways L shape.

Solution

ⓐ 8 units
ⓑ 3 sq. units

Five squares are shown, in a T-shape. There are three squares across the top and three squares down.

Use Properties of Rectangles

In the following exercises, find the ⓐ perimeter ⓑ area of each rectangle

The length of a rectangle is $42$ meters and the width is $28$ meters.

Solution

ⓐ 140 m
ⓑ 1176 sq. m

The length of a rectangle is $36$ feet and the width is $19$ feet.

A sidewalk in front of Kathy’s house is in the shape of a rectangle $4$ feet wide by $45$ feet long.

Solution

ⓐ 98 ft.
ⓑ 180 sq. ft.

A rectangular room is $16$ feet wide by $12$ feet long.

In the following exercises, solve.

Find the length of a rectangle with perimeter of $220$ centimeters and width of $85$ centimeters.

Solution

25 cm

Find the width of a rectangle with perimeter $39$ and length $11 .$

The area of a rectangle is $2356$ square meters. The length is $38$ meters. What is the width?

Solution

62 m

The width of a rectangle is $45$ centimeters. The area is $2700$ square centimeters. What is the length?

The length of a rectangle is $12$ centimeters more than the width. The perimeter is $74$ centimeters. Find the length and the width.

Solution

24.5 cm., 12.5 cm.

The width of a rectangle is $3$ more than twice the length. The perimeter is $96$ inches. Find the length and the width.

Use Properties of Triangles

In the following exercises, solve using the properties of triangles.

Find the area of a triangle with base $18$ inches and height $15$ inches.

Solution

135 sq. in.

Find the area of a triangle with base $33$ centimeters and height $21$ centimeters.

A triangular road sign has base $30$ inches and height $40$ inches. What is its area?

Solution

600 sq. in.

If a triangular courtyard has sides $9$ feet and $12$ feet and the perimeter is $32$ feet, how long is the third side?

A tile in the shape of an isosceles triangle has a base of $6$ inches. If the perimeter is $20$ inches, find the length of each of the other sides.

Solution

7 in., 7 in.

Find the length of each side of an equilateral triangle with perimeter of $81$ yards.

The perimeter of a triangle is $59$ feet. One side of the triangle is $3$ feet longer than the shortest side. The third side is $5$ feet longer than the shortest side. Find the length of each side.

Solution

17 ft., 20 ft., 22 ft.

One side of a triangle is three times the smallest side. The third side is $9$ feet more than the shortest side. The perimeter is $39$ feet. Find the lengths of all three sides.

Use Properties of Trapezoids

In the following exercises, solve using the properties of trapezoids.

The height of a trapezoid is $8$ feet and the bases are $11$ and $14$ feet. What is the area?

Solution

100 sq. ft.

The height of a trapezoid is $5$ yards and the bases are $7$ and $10$ yards. What is the area?

Find the area of the trapezoid with height $25$ meters and bases $32.5$ and $21.5$ meters.

Solution

675 sq. m

A flag is shaped like a trapezoid with height $62$ centimeters and the bases are $91.5$ and $78.1$ centimeters. What is the area of the flag?

Solve Geometry Applications: Circles and Irregular Figures

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

3

meters. Find the

ⓐ circumference
ⓑ area of the mosaic

Solution

ⓐ 18.84 m
ⓑ 28.26 sq. m

A circular fountain has radius

8

feet. Find the

ⓐ circumference
ⓑ area of the fountain

Find the diameter of a circle with circumference $150.72$ inches.

Solution

48 in.

Find the radius of a circle with circumference $345.4$ centimeters

Find the Area of Irregular Figures

In the following exercises, find the area of each shaded region.

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.

Solution

30 sq. units

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.

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.

Solution

300 sq. units

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.

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.

Solution

199.25 sq. units

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.

Solve Geometry Applications: Volume and Surface Area

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 $14$ centimeters, width $4.5$ centimeters, and height $10$ centimeters

Solution

ⓐ 630 cu. cm
ⓑ 496 sq. cm

a cube with sides that are $3$ feet long

a cube of tofu with sides $2.5$ inches

Solution

ⓐ 15.625 cu. in.
ⓑ 37.5 sq. in.

a rectangular carton with length $32$ inches, width $18$ inches, and height $10$ inches

Find Volume and Surface Area of Spheres

In the following exercises, find the

ⓐ volume
ⓑ surface area of the sphere.

a sphere with radius $4$ yards

Solution

ⓐ 267.95 cu. yd.
ⓑ 200.96 sq. yd.

a sphere with radius $12$ meters

a baseball with radius $1.45$ inches

Solution

ⓐ 12.76 cu. in.
ⓑ 26.41 sq. in.

a soccer ball with radius $22$ centimeters

Find Volume and Surface Area of Cylinders

In the following exercises, find the

ⓐ volume
ⓑ surface area of the cylinder

a cylinder with radius $2$ yards and height $6$ yards

Solution

ⓐ 75.36 cu. yd.
ⓑ 100.48 sq. yd.

a cylinder with diameter $18$ inches and height $40$ inches

a juice can with diameter $8$ centimeters and height $15$ centimeters

Solution

ⓐ 753.6 cu. cm
ⓑ 477.28 sq. cm

a cylindrical pylon with diameter $0.8$ feet and height $2.5$ feet

Find Volume of Cones

In the following exercises, find the volume of the cone.

a cone with height $5$ meters and radius $1$ meter

Solution

5.233 cu. m

a cone with height $24$ feet and radius $8$ feet

a cone-shaped water cup with diameter $2.6$ inches and height $2.6$ inches

Solution

4.599 cu. in.

a cone-shaped pile of gravel with diameter $6$ yards and height $5$ yards

Solve a Formula for a Specific Variable

Use the Distance, Rate, and Time Formula

In the following exercises, solve using the formula for distance, rate, and time.

A plane flew $4$ hours at $380$ miles per hour. What distance was covered?

Solution

1520 miles

Gus rode his bike for $1 \frac{1}{2}$ hours at $8$ miles per hour. How far did he ride?

Jack is driving from Bangor to Portland at a rate of $68$ miles per hour. The distance is $107$ miles. To the nearest tenth of an hour, how long will the trip take?

Solution

1.6 hours

Jasmine took the bus from Pittsburgh to Philadelphia. The distance is $305$ miles and the trip took $5$ hours. What was the speed of the bus?

Solve a Formula for a Specific Variable

In the following exercises, use the formula $d = r t .$

Solve for $t :$

ⓐ when $d = 403$ and $r = 65$
ⓑ in general

Solution

ⓐ $t = 6.2$
ⓑ $t = \frac{d}{r}$

Solve for $r :$

ⓐ when $d = 750$ and $t = 15$
ⓑ in general

In the following exercises, use the formula $A = \frac{1}{2} b h .$

Solve for $b :$

ⓐ when $A = 416$ and $h = 32$
ⓑ in general

Solution

ⓐ $b = 26$
ⓑ $b = \frac{2 A}{h}$

Solve for $h :$

ⓐ when $A = 48$ and $b = 8$
ⓑ in general

In the following exercises, use the formula $I = P r t .$

Solve for the principal, $P,$ for:

ⓐ $I = $720$ , $r = 4%$ , $t = 3 years$
ⓑ in general

Solution

ⓐ $P = $6000$
ⓑ $P = \frac{I}{(r \cdot t)}$

Solve for the time,

t

for:

ⓐ $I = $3630$ , $P = $11,000$ , $r = 5.5 %$
ⓑ in general

In the following exercises, solve.

Solve the formula

6 x + 5 y = 20

for

y :

ⓐ when $x = 0$
ⓑ in general

Solution

ⓐ $y = 4$
ⓑ $y = \frac{20 - 6 x}{5}$

Solve the formula

2 x + y = 15

for

y :

ⓐ when $x = −5$
ⓑ in general

Solve $a + b = 90$ for $a .$

Solution

a = 90 − b

Solve $180 = a + b + c$ for $a .$

Solve the formula $4 x + y = 17$ for $y .$

Solution

y = 17 − 4x

Solve the formula $- 3 x + y = −6$ for $y .$

Solve the formula $P = 2 L + 2 W$ for $W .$

Solution

$W = \frac{P - 2 L}{2}$

Solve the formula $V = L W H$ for $H .$

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 $12$ children, what is the total number of people on the hike?

The sum of $13$ and twice a number is $- 19 .$ Find the number.

Solution

−16

One number is $3$ less than another number. Their sum is $65 .$ Find the numbers.

Bonita has $$2.95$ in dimes and quarters in her pocket. If she has $5$ more dimes than quarters, how many of each coin does she have?

Solution

7 quarters, 12 dimes

At a concert, $$1600$ in tickets were sold. Adult tickets were $$9$ each and children’s tickets were $$4$ each. If the number of adult tickets was $30$ fewer than twice the number of children’s tickets, how many of each kind were sold?

Find the complement of a $52°$ angle.

Solution

38°

The measure of one angle of a triangle is twice the measure of the smallest angle. The measure of the third angle is $14$ more than the measure of the smallest angle. Find the measures of all three angles.

The perimeter of an equilateral triangle is $145$ feet. Find the length of each side.

Solution

48.3

$Δ A B C$ is similar to $Δ X Y Z .$ Find the length of side $c .$

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.

Find the length of the missing side. Round to the nearest tenth, if necessary.

A right triangle is shown. The height is labeled 24 and the hypotenuse is labeled 26.

Solution

Find the length of the missing side. Round to the nearest tenth, if necessary.

A right triangle is shown. The base is labeled 6 and the height is labeled 9.

A baseball diamond is shaped like a square with sides $90$ feet long. How far is it from home plate to second base, as shown?

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.

Solution

127.3 ft

The length of a rectangle is $2$ feet more than five times the width. The perimeter is $40$ feet. Find the dimensions of the rectangle.

A triangular poster has base $80$ centimeters and height $55$ centimeters. Find the area of the poster.

Solution

2200 square centimeters

A trapezoid has height $14$ inches and bases $20$ inches and $23$ inches. Find the area of the trapezoid.

A circular pool has diameter $90$ inches. What is its circumference? Round to the nearest tenth.

Solution

282.6 inches

Find the area of the shaded region. Round to the nearest tenth.

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.

Find the volume of a rectangular room with width $12$ feet, length $15$ feet, and height $8$ feet.

Solution

1440

A coffee can is shaped like a cylinder with height $7$ inches and radius $5$ inches. Find (a) the surface area and (b) the volume of the can. Round to the nearest tenth.

A traffic cone has height $75$ centimeters. The radius of the base is $20$ centimeters. Find the volume of the cone. Round to the nearest tenth.

Solution

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 $63$ miles per hour and the trip took $4$ hours. What was the distance?

The Catalina Express takes $1 \frac{1}{2}$ hours to travel from Long Beach to Catalina Island, a distance of $22$ miles. To the nearest tenth, what is the speed of the boat?

Solution

14.7 miles per hour

Use the formula

I = P r t

to solve for the principal,

P,

for:

ⓐ $I = $1380, r = 5%, t = 3$ years
ⓑ in general

Solve the formula

A = \frac{1}{2} b h

for

h :

ⓐ when $A = 1716$ and $b = 66$
ⓑ in general

Solution

ⓐ $height = 52$
ⓑ $h = \frac{2 A}{b}$

Solve $x + 5 y = 14$ for $y .$