Section 2.3
2.3
Formulas and Problem Solving 69
Formulas and Problem Solving
OBJECTIVE
OBJECTIVES
1 Solve a Formula for a Specified
Variable.
2 Use Formulas to Solve
1
Solving a Formula for a Specified Variable
Solving problems that we encounter in the real world sometimes requires us to express
relationships among measured quantities. An equation that describes a known relationship among quantities such as distance, time, volume, weight, money, and gravity is
called a formula. Some examples of formulas are
Problems.
Formula
I = PRT
A = lw
d = rt
C = 2pr
V = lwh
Meaning
Interest = principal # rate # time
Area of a rectangle = length # width
Distance = rate # time
Circumference of a circle = 2 # p # radius
Volume of a rectangular solid = length # width # height
Other formulas are listed in the front cover of this text. Notice that the formula for the
volume of a rectangular solid V = lwh is solved for V since V is by itself on one side
of the equation with no V’s on the other side of the equation. Suppose that the volume
of a rectangular solid is known as well as its width and its length, and we wish to find
its height. One way to find its height is to begin by solving the formula V = lwh for h.
EXAMPLE 1
Solve: V = lwh for h.
Solution To solve V = lwh for h, isolate h on one side of the equation. To do so,
divide both sides of the equation by lw.
V = lwh
V
lw h
=
lw
lw
V
V
= h or h =
lw
lw
Divide both sides by lw.
Simplify.
Thus we see that to find the height of a rectangular solid, we divide its volume by the
product of its length and its width.
PRACTICE
1
Solve: I = PRT for T.
The following steps may be used to solve formulas and equations in general for a
specified variable.
Solving Equations for a Specified Variable
Step 1.
Step 2.
Step 3.
Step 4.
Step 5.
Clear the equation of fractions by multiplying each side of the equation
by the least common denominator.
Use the distributive property to remove grouping symbols such as
parentheses.
Combine like terms on each side of the equation.
Use the addition property of equality to rewrite the equation as an
equivalent equation with terms containing the specified variable on one
side and all other terms on the other side.
Use the distributive property and the multiplication property of equality
to isolate the specified variable.
70 CHAPTER 2
Equations, Inequalities, and Problem Solving
EXAMPLE 2
Solve: 3y - 2x = 7 for y.
Solution This is a linear equation in two variables. Often an equation such as this is
solved for y to reveal some properties about the graph of this equation, which we will
learn more about in Chapter 3. Since there are no fractions or grouping symbols, we
begin with Step 4 and isolate the term containing the specified variable y by adding 2x
to both sides of the equation.
3y - 2x = 7
3y - 2x + 2x = 7 + 2x
3y = 7 + 2x
Add 2x to both sides.
To solve for y, divide both sides by 3.
3y
7 + 2x
=
3
3
y =
Divide both sides by 3.
2x + 7
3
or
y =
2x
7
+
3
3
PRACTICE
2
Solve: 7x - 2y = 5 for y.
EXAMPLE 3
Solve: A =
1
1B + b2h for b.
2
Solution Since this formula for finding the area of a trapezoid contains fractions, we
begin by multiplying both sides of the equation by the LCD 2.
b
Helpful Hint
Remember that we may get
the specified variable alone on
either side of the equation.
h
B
1
1B + b2h
2
1
2 # A = 2 # 1B + b2h
2
2A = 1B + b2h
A =
Multiply both sides by 2.
Simplify.
Next, use the distributive property and remove parentheses.
2A = 1B + b2h
2A = Bh + bh
2A - Bh = bh
2A - Bh
bh
=
h
h
2A - Bh
2A - Bh
= b or b =
h
h
Apply the distributive property.
Isolate the term containing b by
subtracting Bh from both sides.
Divide both sides by h.
PRACTICE
3
Solve: A = P + Prt for r.
OBJECTIVE
2
Using Formulas to Solve Problems
In this section, we also solve problems that can be modeled by known formulas. We
use the same problem-solving steps that were introduced in the previous section.
Section 2.3
Formulas and Problem Solving 71
Formulas are very useful in problem solving. For example, the compound interest
formula
A = Pa 1 +
r nt
b
n
is used by banks to compute the amount A in an account that pays compound interest.
The variable P represents the principal or amount invested in the account, r is the annual rate of interest, t is the time in years, and n is the number of times compounded
per year.
EXAMPLE 4
Finding the Amount in a Savings Account
Karen Estes just received an inheritance of $10,000 and plans to place all the money in
a savings account that pays 5% compounded quarterly to help her son go to college in
3 years. How much money will be in the account in 3 years?
Solution
1. UNDERSTAND. Read and reread the problem. The appropriate formula to solve
this problem is the compound interest formula
A = Pa 1 +
r nt
b
n
Make sure that you understand the meaning of all the variables in this formula.
A
P
t
r
n
=
=
=
=
=
amount in the account after t years
principal or amount invested
time in years
annual rate of interest
number of times compounded per year
2. TRANSLATE. Use the compound interest formula and let P = +10,000,
r = 5, = 0.05, t = 3 years , and n = 4 since the account is compounded quarterly, or 4 times a year.
Formula:
A = Pa 1 +
r nt
b
n
#
0.05 4 3
Substitute: A = 10,000 a 1 +
b
4
3. SOLVE. We simplify the right side of the equation.
#
0.05 4 3
b
4
0.05
A = 10,00011.01252 12
Simplify 1 +
and write 4 # 3 as 12.
4
A ⬇ 10,00011.1607545182 Approximate 11.01252 12.
A ⬇ 11,607.55
Multiply and round to two decimal places.
A = 10,000 a 1 +
4. INTERPRET.
Check: Repeat your calculations to make sure that no error was made. Notice that
$11,607.55 is a reasonable amount to have in the account after 3 years.
State:
In 3 years, the account will contain $11,607.55.
PRACTICE
4
Russ placed $8000 into his credit union account paying 6% compounded
semiannually (twice a year). How much will be in Russ’s account in 4 years?
72 CHAPTER 2
Equations, Inequalities, and Problem Solving
Graphing Calculator Explorations
To solve Example 4, we approximated the expression
#
10,000a 1 +
0.05 4 3
b
4
Use the keystrokes shown in the accompanying calculator screen to evaluate this expression using a
graphing calculator. Notice the use of parentheses.
EXAMPLE 5
Finding Cycling Time
The fastest average speed by a cyclist across the continental United States is 15.4 mph,
by Pete Penseyres. If he traveled a total distance of about 3107.5 miles at this speed,
find his time cycling. Write the time in days, hours, and minutes. (Source: The Guinness
Book of World Records)
Solution
1. UNDERSTAND. Read and reread the problem. The appropriate formula is the
distance formula
d = rt
where
d = distance traveled r = rate and t = time
2. TRANSLATE. Use the distance formula and let d = 3107.5 miles and r = 15.4 mph .
d = rt
3107.5 = 15.4t
3. SOLVE.
3107.5
15.4t
=
15.4
15.4
201.79 ⬇ t
Divide both sides by 15.4.
The time is approximately 201.79 hours. Since there are 24 hours in a day, we divide
201.79 by 24 and find that the time is approximately 8.41 days. Now, let’s convert the
decimal part of 8.41 days back to hours. To do this, multiply 0.41 by 24 and the result
is 9.84 hours. Next, we convert the decimal part of 9.84 hours to minutes by multiplying by 60 since there are 60 minutes in an hour. We have 0.84 # 60 ⬇ 50 minutes
rounded to the nearest whole. The time is then approximately
8 days, 9 hours, 50 minutes.
4. INTERPRET.
Check: Repeat your calculations to make sure that an error was not made.
State:
Pete Penseyres’s cycling time was approximately 8 days, 9 hours, 50 minutes.
PRACTICE
5
Nearly 4800 cyclists from 36 U.S. states and 6 countries rode in the PanMassachusetts Challenge recently to raise money for cancer research and treatment.
If the riders of a certain team traveled their 192-mile route at an average speed of
7.5 miles per hour, find the time they spent cycling. Write the answer in hours and
minutes.
Section 2.3
Formulas and Problem Solving 73
Vocabulary, Readiness & Video Check
Solve each equation for the specified variable. See Examples 1 through 3.
1. 2x + y = 5 for y
2. 7x - y = 3 for y
3. a - 5b = 8 for a
4. 7r + s = 10 for s
5. 5j + k - h = 6 for k
6. w - 4y + z = 0 for z
Martin-Gay Interactive Videos
Watch the section lecture video and answer the following questions.
OBJECTIVE
1
OBJECTIVE
2
7. Based on the lecture before Example 1, what two things does solving
an equation for a specific variable mean?
8. As the solution is checked at the end of Example 3, why do you think
it is mentioned to be especially careful that you use the correct formula
when solving problems?
See Video 2.3
2.3
Exercise Set
Solve each equation for the specified variable. See Examples 1–3.
1. d = rt; for t
2. W = gh; for g
3. I = PRT; for R
4. V = lwh; for l
5. 9x - 4y = 16; for y
6. 2x + 3y = 17; for y
7. P = 2L + 2W; for W
21. N = 3st 4 - 5sv; for v
22. L = a + 1n - 12d; for d
23. S = 2LW + 2LH + 2WH; for H
24. T = 3vs - 4ws + 5vw; for v
In this exercise set, round all dollar amounts to two decimal places.
Solve. See Example 4.
25. Complete the table and find the balance A if $3500 is
invested at an annual percentage rate of 3% for 10 years and
compounded n times a year.
8. A = 3M - 2N; for N
n
9. J = AC - 3; for A
A
10. y = mx + b; for x
11. W = gh - 3gt 2; for g
n
13. T = C12 + AB2; for B
A
15. C = 2pr; for r
16. S = 2pr 2 + 2prh; for h
17. E = I1r + R2; for r
18. A = P11 + rt2; for t
n
1a + L2; for L
2
5
20. C = 1F - 322; for F
9
19. s =
2
4
12
365
26. Complete the table and find the balance A if $5000 is
invested at an annual percentage rate of 6% for 15 years and
compounded n times a year.
12. A = Prt + P; for P
14. A = 5H1b + B2; for b
1
1
2
4
12
365
27. A principal of $6000 is invested in an account paying an
annual percentage rate of 4%. Find the amount in the
account after 5 years if the account is compounded
a. semiannually
b. quarterly
c. monthly
28. A principal of $25,000 is invested in an account paying
an annual percentage rate of 5%. Find the amount in the
account after 2 years if the account is compounded
a. semiannually
b. quarterly
c. monthly
74 CHAPTER 2
Equations, Inequalities, and Problem Solving
MIXED PRACTICE
Solve. For Exercises 29 and 30, the solutions have been started for
you. Round all dollar amounts to two decimal places. See Examples 4 and 5.
29. Omaha, Nebraska, is about 90 miles from Lincoln, Nebraska. Irania Schmidt must go to the law library in Lincoln to get a document for the law firm she works for. Find
how long it takes her to drive round-trip if she averages
50 mph.
34. One-foot-square ceiling tiles are sold in packages of 50. Find
how many packages must be bought for a rectangular ceiling
18 feet by 12 feet.
35. If the area of a triangular kite is 18 square feet and its base is
4 feet, find the height of the kite.
4 ft
height
Start the solution:
1. UNDERSTAND the problem. Reread it as many times
as needed.
2. TRANSLATE into an equation. (Fill in the blanks below.)
Here, we simply use the formula d = r # t.
distance
(round-trip)
T
equals
T
=
rate or
speed
#
T
T
#
time
T
t
Finish with:
3. SOLVE and 4. INTERPRET
1
30. It took the Selby family 5 hours round-trip to drive from
2
their house to their beach house 154 miles away. Find their
average speed.
Start the solution:
1. UNDERSTAND the problem. Reread it as many times
as needed.
2. TRANSLATE into an equation. (Fill in the blanks below.)
Here, we simply use the formula d = r # t.
distance
(round-trip)
T
equals
rate or
speed
#
time
T
=
T
r
T
T
#
36. Bailey, Ethan, Avery, Mia, and Madison would like to go to
Disneyland in 3 years. Their total cost should be $4500. If
each invests $800 in a savings account paying 5.5% interest compounded semiannually, will they have enough in
3 years?
37. A gallon of latex paint can cover 500 square feet. Find how
many gallon containers of paint should be bought to paint
two coats on each wall of a rectangular room whose dimensions are 14 feet by 16 feet (assume 8-foot ceilings).
38. A gallon of enamel paint can cover 300 square feet. Find
how many gallon containers of paint should be bought to
paint three coats on a wall measuring 21 feet by 8 feet.
To prepare for Exercises 43 and 44, use the volume formulas below
to solve Exercises 39–42. Remember, volume is measured in cubic
units.
4 3
Cylinder: V = pr 2h
Sphere: V =
pr
3
39. The cylinder below has an exact volume of 980p cubic
meters. Find its height.
7m
Finish with:
height
3. SOLVE and 4. INTERPRET
31. The day’s high temperature in Phoenix, Arizona, was
recorded as 104°F. Write 104°F as degrees Celsius. [Use the
5
formula C = 1F - 322.]
9
32. The annual low temperature in Nome, Alaska, was recorded
as - 15C . Write -15C as degrees Fahrenheit. [Use the
9
formula F = C + 32.]
5
33. A package of floor tiles contains 24 one-foot-square tiles.
Find how many packages should be bought to cover a square
ballroom floor whose side measures 64 feet.
40. The battery below is in the shape of a cylinder and has an
exact volume of 825p cubic millimeters. Find its height.
r 5 mm
height
41. The steel ball below is in the shape of a sphere and has a
diameter of 12 millimeters.
d 12 mm
64 ft
64 ft
a. Find the exact volume of the sphere.
b. Find a 2-decimal-place approximation for the volume.
Section 2.3
42. The spherical ball below has a diameter of 18 centimeters.
d 18 cm
Formulas and Problem Solving 75
46. In 1945, Arthur C. Clarke, a scientist and science-fiction
writer, predicted that an artificial satellite placed at a height
of 22,248 miles directly above the equator would orbit the
globe at the same speed with which the earth was rotating.
This belt along the equator is known as the Clarke belt.
Use the formula for circumference of a circle and find the
“length” of the Clarke belt. (Hint: Recall that the radius of
the earth is approximately 4000 miles. Round to the nearest
whole mile.)
a. Find the exact volume of the ball.
b. Find a 2-decimal-place approximation of the volume.
43. A portion of the external tank of the Space Shuttle
Endeavour was a liquid hydrogen tank. If the ends of the
tank are hemispheres, find the volume of the tank. To do so,
answer parts a through c. (Note: Endeavour completed its
last mission in 2011.)
4.2 m
21.2 m
4.2 m
Cylinder
Sphere
a. Find the volume of the cylinder shown. Round to 2 decimal places.
b. Find the volume of the sphere shown. Round to 2 decimal places.
c. Add the results of parts a and b. This sum is the approximate volume of the tank.
22,248 mi
47. The deepest hole in the ocean floor is beneath the Pacific
Ocean and is called Hole 504B. It is located off the coast
of Ecuador. Scientists are drilling it to learn more about
the earth’s history. Currently, the hole is in the shape of a
cylinder whose volume is approximately 3800 cubic feet and
whose length is 1.3 miles. Find the radius of the hole to the
nearest hundredth of a foot. (Hint: Make sure the same units
of measurement are used.)
48. The deepest man-made hole is called the Kola Superdeep
Borehole. It is approximately 8 miles deep and is located
near a small Russian town in the Arctic Circle. If it takes
7.5 hours to remove the drill from the bottom of the hole,
find the rate that the drill can be retrieved in feet per second. Round to the nearest tenth. (Hint: Write 8 miles as feet,
7.5 hours as seconds, then use the formula d = rt.)
49. Eartha is the world’s largest globe. It is located at the headquarters of DeLorme, a mapmaking company in Yarmouth,
Maine. Eartha is 41.125 feet in diameter. Find its exact
circumference (distance around) and then approximate its
circumference using 3.14 for p . (Source: DeLorme)
44. A vitamin is in the shape of a cylinder with a hemisphere at
each end, as shown. Use the art in Exercise 43 to help find
the volume of the vitamin.
4 mm
15 mm
a. Find the volume of the cylinder part. Round to two decimal places.
b. Find the volume of the sphere formed by joining the two
hemispherical ends. Round to two decimal places.
c. Add the results of parts a and b to find the approximate
volume of the vitamin.
45. Amelia Earhart was the first woman to fly solo nonstop coast
to coast, setting the women’s nonstop transcontinental speed
record. She traveled 2447.8 miles in 19 hours 5 minutes. Find
the average speed of her flight in miles per hour. (Change
19 hours 5 minutes into hours and use the formula d = rt.)
Round to the nearest tenth of a mile per hour.
50. Eartha is in the shape of a sphere. Its radius is about 20.6 feet.
Approximate its volume to the nearest cubic foot. (Source:
DeLorme)
51. Find how much interest $10,000 earns in 2 years in a
certificate of deposit paying 8.5% interest compounded
quarterly.
52. Find how long it takes Mark to drive 135 miles on I-10 if
he merges onto I-10 at 10 a.m. and drives nonstop with his
cruise control set on 60 mph.
76 CHAPTER 2
Equations, Inequalities, and Problem Solving
The calorie count of a serving of food can be computed based
on its composition of carbohydrate, fat, and protein. The calorie
count C for a serving of food can be computed using the formula
C = 4h + 9f + 4p, where h is the number of grams of carbohydrate contained in the serving, f is the number of grams of fat
contained in the serving, and p is the number of grams of protein
contained in the serving.
53. Solve this formula for f, the number of grams of fat contained in a serving of food.
54. Solve this formula for h, the number of grams of carbohydrate contained in a serving of food.
55. A serving of cashews contains 14 grams of fat, 7 grams of
carbohydrate, and 6 grams of protein. How many calories
are in this serving of cashews?
56. A serving of chocolate candies contains 9 grams of fat,
30 grams of carbohydrate, and 2 grams of protein. How many
calories are in this serving of chocolate candies?
57. A serving of raisins contains 130 calories and 31 grams of
carbohydrate. If raisins are a fat-free food, how much protein is provided by this serving of raisins?
58. A serving of yogurt contains 120 calories, 21 grams of carbohydrate, and 5 grams of protein. How much fat is provided by this
serving of yogurt? Round to the nearest tenth of a gram.
REVIEW AND PREVIEW
Determine which numbers in the set 5 - 3 , - 2 , - 1 , 0 , 1 , 2 , 3 6 are
solutions of each inequality. See Sections 1.4 and 2.1.
59. x 6 0
60. x 7 1
61. x + 5 … 6
62. x - 3 Ú - 7
63. In your own words, explain what real numbers are solutions
of x 6 0 .
64. In your own words, explain what real numbers are solutions
of x 7 1 .
Planet
Miles from the Sun
70.
Uranus
1783 million
71.
Neptune
2793 million
72.
Pluto
(dwarf planet)
3670 million
73. To borrow money at a rate of r, which loan plan should you
choose—one compounding 4 times a year or 12 times a
year? Explain your choice.
74. If you are investing money in a savings account paying a
rate of r, which account should you choose—an account
compounded 4 times a year or 12 times a year? Explain your
choice.
75. To solve the formula W = gh - 3gt 2 for g, explain why it is
a good idea to factor g first from the terms on the right side
of the equation. Then perform this step and solve for g.
76. An orbit such as Clarke’s belt in Exercise 46 is called a
geostationary orbit. In your own words, why do you think
that communications satellites are placed in geostationary
orbits?
The measure of the chance or likelihood of an event occurring is
its probability. A formula basic to the study of probability is the
formula for the probability of an event when all the outcomes are
equally likely. This formula is
number of ways that
the event can occur
Probability of an event =
number of possible
outcomes
For example, to find the probability that a single spin on the spinner below will result in red, notice first that the spinner is divided
into 8 parts, so there are 8 possible outcomes. Next, notice that
there is only one sector of the spinner colored red, so the number
of ways that the spinner can land on red is 1. Then this probability
denoted by P(red) is
CONCEPT EXTENSIONS
P(red) Ω
Solar system distances are so great that units other than miles or
kilometers are often used. For example, the astronomical unit (AU)
is the average distance between Earth and the sun, or 92,900,000
miles. Use this information to convert each planet’s distance in
miles from the sun to astronomical units. Round to three decimal
places. The planet Mercury’s AU from the sun has been completed
for you. (Source: National Space Science Data Center)
Planet
Mercury
AU from the Sun
Miles from the Sun
AU from the Sun
36 million
0.388
65.
Venus
67.2 million
66.
Earth
92.9 million
67.
Mars
141.5 million
68.
Jupiter
483.3 million
69.
Saturn
886.1 million
Find each probability in simplest form.
77. P(green)
78. P(yellow)
79. P(black)
80. P(blue)
81. P(green or blue)
82. P(black or yellow)
83. P(red, green, or black)
84. P(yellow, blue, or black)
85. P(white)
86. P(red, yellow, green, blue, or black)
87. From the previous probability formula, what do you think
is always the probability of an event that is impossible to
occur?
88. What do you think is always the probability of an event that
is sure to occur?