Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach
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Dividing both sides by 4 and applying the quadratic formula yields
x
2
32x 165 0
x
4.5183
x
32
2
1684
or
36.5183.
Only the positive solution makes sense in the context of this problem. The walk
should be approximately 4.5 feet wide. ■
The basic formula for problems involving distance and a uniform rate of
speed is
Distance rate time.
For instance, if you drive at a rate of 55 mph for 2 hours, you travel a distance
of 55
2 110 miles.
EXAMPLE 7
A stone is dropped from the top of a cliff. Five seconds later, the sound of the
stone hitting the ground is heard. Use the fact that the speed of sound is 1100 feet
per second and that the distance traveled by a falling object in t seconds is 16t
2
feet to determine the height of the cliff.
SOLUTION If t is the time it takes the rock to fall to the ground, then the height
of the cliff is 16t
2
feet. The total time for the rock to fall and the sound of its hit-
ting ground to return to the top of the cliff is 5 seconds. So the sound must take
5 t seconds to return to the top, as indicated in Figure 2–42. Therefore,
Height of cliff
16t
2
Rate
Time
16t
2
1100(5 t)
16t
2
5500 1100t
16t
2
1100t 5500 0.
The quadratic formula and a calculator show that
t
1100
1,562,0
00
32
1100
(1100)
2
4
16
(550
0)
2
16
Distance sound
travels upward
in 5 t seconds
Distance rock falls
in t seconds
32
(32)
2
4
1
(165
)
2
1
106 CHAPTER 2 Graphs and Technology
4.6812
or
73.4312.
Rock
falls
for t sec
Thud!
Sound
rises for
5 − t secs
Figure 2–42
Since the time must be a number between 0 and 5, we see that t 4.6812 seconds.
Therefore the height of the cliff is
16t
2
16(4.6812)
2
350.6 feet. ■
EXAMPLE 8
A pilot wants to make the 840-mile round trip from Cleveland to Peoria and back
in 5 hours flying time. Going to Peoria, there will be a headwind of 30 mph, that
is, a wind opposite to the direction the plane is flying. It is estimated that on the
return trip to Cleveland, there will be a 40-mph tailwind (in the direction the plane
is flying). At what constant speed should the plane be flown?
SOLUTION Let x be the engine speed of the plane. On the trip to Peoria, the
actual speed will be x 30 mph because of the headwind.
x mph
Plane
Wind
30 mph
Actual speed
x 30 mph
The distance from Cleveland to Peoria is 420 miles (half the round trip). So we
have
Rate
Time Distance
Time
Di
R
st
a
a
t
n
e
ce
x
4
20
30
.
On the return trip, the actual speed will be x 40 because of the tailwind.
x mph 40 mph
Plane Wind
Actual speed
x 40 mph
So we have
Rate
Time Distance
Time
Di
R
st
a
a
t
n
e
ce
x
4
20
40
.
Therefore,
5
5
x
4
20
30
x
4
20
40
.
Multiplying both sides by the common denominator (x 30)(x 40) and sim-
plifying, we have
Time from Peoria
to Cleveland
Time from Cleveland
to Peoria
SECTION 2.3 Applications of Equations 107
5(x 30)(x 40)
x
4
20
30
(x 30)(x 40)
x
4
20
40
(x 30)(x 40)
5(x 30)(x 40) 420(x 40) 420(x 30)
(x 30)(x 40) 84(x 40) 84(x 30)
x
2
10x 1200 84x 3360 84x 2520
x
2
158x 2040 0
(x 170)(x 12) 0
x 170 0orx 12 0
x 170 x 12.
Obviously, the negative solution doesn’t apply. Since we multiplied both sides by a
quantity involving the variable, we must check that 170 actually is a solution of the
original equation. It is, so the plane should be flown at a speed of 170 mph. ■
EXAMPLE 9
A rectangular box with a square base and no top is to have a volume of
20,000 cubic centimeters. If the surface area of the box is 4000 square centimeters
and the box is required to be higher than it is wide, what are its dimensions?
SOLUTION This is essentially Example 3 with an extra condition (the box
must be higher than it is wide). If the length, width, and height of the box are x, x,
and h as shown in Figure 2–43, then as we saw in Example 3,
x
2
80,
x
000
4000 and h
20
x
,0
2
00
.
Multiplying both sides of the first equation by x (which is nonzero because it is a
length), we obtain
x
3
80,000 4000x
x
3
4000x 80,000 0.
This equation can be solved in more than one way.
Graphical: The graph of y x
3
4000x 80,000 in Figure 2–44 shows that
the equation has three solutions. The negative solution does not apply here,
nor does the one near 50 (because a value of x in that range makes the cor-
responding height h 20,000/x
2
smaller than the width x, as indicated in
Figure 2–45). The third solution, x 23.069, is easily found (Figure 2–46).
108 CHAPTER 2 Graphs and Technology
h
x
x
Figure 2–43
200,000
−100,000
−80 80
Figure 2–44
Figure 2–45
200,000
−100,000
−80 80
Figure 2–46
Numerical: The polynomial solver on a TI-86 or the “fsolve” command on
Maple produces the three solutions (Figure 2–47). As above, only one of
them is applicable here, namely, x 23.069.
Figure 2–47
Therefore, the height is
h
20
x
,0
2
00
2
2
3
0
.
,
0
0
6
0
9
0
2
37.581.
So the approximate dimensions of the box are 23.069 by 23.069 by 37.581 cm. ■
EXAMPLE 10
A box (with no top) of volume 1000 cubic inches is to be made from a 22
30 inch sheet of cardboard by cutting squares of equal size from each corner
and folding up the flaps, as shown in Figure 2–48. If the box must be at least
4 inches high, what size square should be cut from each corner?
Figure 2–48
SOLUTION Let x denote the length of the side of the square to be cut from
each corner. The dashed rectangle in Figure 2–48 is the bottom of the box. Its
length is 30 2x as shown in the figure. Similarly, the width of the box will be
22 2x, and its height will be x inches. Therefore,
Length Width Height Volume of box
(30 2x)
(22 2x)
x 1000
(660 104x 4x
2
)x 1000
4x
3
104x
2
660x 1000 0.
22
30
30 − 2x
xx
x
x
22 − 2x
> fsolve(x^3 − 4000*x + 80000 = 0, x);
−71.5417889, 23.0693461, 48.4724428
MAPLE
SECTION 2.3 Applications of Equations 109
——
——
——
———
Since the cardboard is 22 inches wide, x must be less than 11 (otherwise, you can’t
cut out two squares of length x). Since x is a length, it is positive. So we need only
find solutions of the equation between 0 and 11. We graph
y 4x
3
104x
2
660x 1000
in a window with 0 x 11 (Figure 2–49). A complete graph isn’t needed here,
only the x-intercepts (solutions). The one between 2 and 3 is not relevant here
because x is the height of the box, which must be at least 4 inches.
110 CHAPTER 2 Graphs and Technology
EXAMPLE 11
Sharon Mahoney was at point A on the bank of a 2.5-kilometer wide river and
traveled to point B, 15 kilometers downstream on the opposite bank. She rowed to
point C and then ran to B, as shown in Figure 2–50. She rowed at a rate of 4 kilo-
meters per hour and ran at 8 kilometers per hour. Her trip took 3 hours. How far
did she run?
15 − x
x
15
2.5
C
d
B
D
A
Figure 2–50
SOLUTION Let x be the distance that Sharon ran from C to B. Using the basic
formula for distance, we have
Rate Time Distance
Time
Di
R
st
a
a
t
n
e
ce
8
x
.
Similarly, the time required to row distance d is
Time
Di
R
st
a
a
t
n
e
ce
d
4
.
Since 15 x is the distance from D to C, the Pythagorean Theorem applied to
right triangle ADC shows that
d
2
(15 x)
2
2.5
2
or, equivalently, d (15
x)
2
6
.25
.
Use a root finder or a polynomial solver on a calculator or computer to find the
solution of the equation between 6 and 7. This is the side x of the square that
should be cut from each corner. Round the value of x to two decimal places, and
find the dimensions of the resulting box.
■
GRAPHING EXPLORATION
10
−10
011
Figure 2–49
Therefore, the total time for the trip is given by
Rowing time Running time
d
4
8
x
8
x
.
If the trip took 3 hours, then we must solve the equation
8
x
3.
(x 1
5)
2
6.25
4
(x 1
5)
2
6.25
4
SECTION 2.3 Applications of Equations 111
This Graphing Exploration shows that Sharon ran approximately 6.74 kilometers
from C to B. ■
EXERCISES 2.3
In Exercises 1–4, a problem situation is given.
(a) Decide what is being asked for, and label the unknown
quantities.
(b) Translate the verbal statements in the problem and the
relationships between the known and unknown quantities
into mathematical language, using a table as in Exam-
ples 1–3 (pages 101–103). The table is provided in Exer-
cises 1 and 2. You need not find an equation to be solved.
1. A student has exam scores of 88, 62, and 79. What score
does he need on the fourth exam to have an average of 80?
Mathematical
English Language Language
Score on fourth exam
Sum of scores on four exams
Average of scores on four exams
2. How many gallons of a 12% salt solution should be com-
bined with 10 gallons of an 18% salt solution to obtain a
16% solution?
Mathematical
English Language Language
Gallons of 12% solution
Total gallons of mixture
Amount of salt in 10 gallons of the
18% solution
Amount of salt in the 12% solution
Amount of salt in the mixture
3. A rectangle has perimeter of 45 centimeters and an area of
112.5 square centimeters. What are its dimensions?
4. A triangle has area 96 square inches, and its height is two-
thirds of its base. What are the base and height of the triangle?
In Exercises 5–8, set up the problem by labeling the unknowns,
translating the given information into mathematical language,
and finding an equation that will produce the solution to the
problem. You need not solve this equation.
5. A worker gets an 8% pay raise and now makes $2619 per
month. What was the worker’s old salary?
6. A merchant has 5 pounds of mixed nuts that cost $30. He
wants to add peanuts that cost $1.50 per pound and cashews
that cost $4.50 per pound to obtain 50 pounds of a mixture
that costs $2.90 per pound. How many pounds of peanuts
are needed?
7. The diameter of a circle is 16 cm. By what amount must the
radius be decreased to decrease the area by 48p square
centimeters?
8. A corner lot has dimensions 25 by 40 yards. The city plans
to take a strip of uniform width along the two sides border-
ing the streets to widen these roads. How wide should
the strip be if the remainder of the lot is to have an area of
844 square yards?
In the remaining exercises, solve the applied problems.
9. You have already invested $550 in a stock with an annual
return of 11%. How much of an additional $1100 should be
Using the viewing window with 0 x 15 and 2 y 2, graph
y
8
x
3
and use a root finder to find its x-intercept (the solution of the equation).
(x 1
5)
2
6.25
4
GRAPHING EXPLORATION
invested at 12% and how much at 6% so that the total return
on the entire $1650 is 9%?
10. If you borrow $500 from a credit union at 12% annual
interest and $250 from a bank at 18% annual interest, what
is the effective annual interest rate (that is, what single rate
of interest on $750 would result in the same total amount of
interest)?
11. A radiator contains 8 quarts of fluid, 40% of which is
antifreeze. How much fluid should be drained and replaced
with pure antifreeze so that the new mixture is 60%
antifreeze?
12. A radiator contains 10 quarts of fluid, 30% of which is
antifreeze. How much fluid should be drained and replaced
with pure antifreeze so that the new mixture is 40%
antifreeze?
13. An airplane flew with the wind for 2.5 hours and returned
the same distance against the wind in 3.5 hours. If
the cruising speed of the plane was a constant 360 mph
in air, how fast was the wind blowing?
14. A train leaves New York for Boston, 200 miles away,
at 3:00
P.M. and averages 75 mph. Another train leaves
Boston for New York on an adjacent set of tracks at
5:00
P.M. and averages 45 mph. At what time will the trains
meet?
15. The average of two real numbers is 41.125, and their
product is 1683. What are the numbers? [Hint: See Exam-
ple 1.]
16. A rectangle is twice as long as it is wide. If it has an area
of 24.5 square inches, what are its dimensions? [Hint: See
Example 2.]
17. Two cars leave a gas station at the same time, one traveling
north and the other south. The northbound car travels at
50 mph. After 3 hours, the cars are 345 miles apart. How
fast is the southbound car traveling?
18. A student leaves the university at noon, bicycling south at a
constant rate. At 12:30
P.M., a second student leaves the
same point and heads west, bicycling 7 mph faster than the
first student. At 2:00
P.M., they are 30 miles apart. How fast
is each one going?
19. A 13-foot-long ladder leans on a wall, as shown in
the figure. The bottom of the ladder is 5 feet from the
wall. If the bottom is pulled out 3 feet farther from
the wall, how far does the top of the ladder move down
the wall? [Hint: Draw pictures of the right triangle
formed by the ladder, the ground, and the wall before
and after the ladder is moved. In each case, use the
Pythagorean Theorem to find the distance from the top of
the ladder to the ground.]
20. A 15-foot-long pole leans against a wall. The bottom is
9 feet from the wall. How much farther should the bottom
be pulled away from the wall so that the top moves the same
amount down the wall?
21. A concrete walk of uniform width is to be built around a
circular pool, as shown in the figure. The radius of the
pool is 12 meters, and enough concrete is available to cover
52p square meters. If all the concrete is to be used, how
wide should the walk be?
22. In Example 14 of Section 1.4 (page 63), how many can
openers must be produced to have an average cost per can
opener of $3?
23. Tom drops a rock into a well and 3 seconds later hears
the sound of its splash. How deep is the well? [Make the
same assumptions about falling rocks and sound as in Ex-
ample 7 (page 106).]
Rock falls
Sound rises
12
5 ft
112 CHAPTER 2 Graphs and Technology
24. A group of homeowners are to share equally in the $210
cost of repairing a bus-stop shelter near their homes. At
the last moment, two members of the group decide not to
participate, and this raises the share of each remaining
person by $28. How many people were in the group at the
beginning?
25. Red Riding Hood drives the 432 miles to Grandmother’s
house in 1 hour less than it takes the Wolf to drive the same
route. Her average speed is 6 mph faster than the Wolf’s
average speed. How fast does each drive?
26. To get to work, Sam jogs 3 kilometers to the train and then
rides the remaining 5 kilometers. If the train goes 40 kilo-
meters per hour faster than Sam’s constant rate of jogging
and the entire trip takes 30 minutes, how fast does Sam jog?
27. Suppose that the open-top box being made from a sheet of
cardboard in Example 10 (page 109) may have any height, but
is required to have at least one of its dimensions greater than
18 inches. What size square should be cut from each corner?
28. The dimensions of a rectangular box are consecutive inte-
gers. If the box has volume of 13,800 cubic centimeters,
what are its dimensions?
29. The surface area S of the right circular cone in the figure
below is given by S pr
r
2
h
2
. What radius should
be used to produce a cone of height 5 inches and surface
area 100 square inches?
30. What is the radius of the base of a cone whose surface area
is 18p square centimeters and whose height is 4 cm?
31. Find the radius of the base of a conical container whose
height is 1/3 of the radius and whose volume is 180 cubic
inches. [Note: The volume of a cone of radius r and height
h is pr
2
h/3.]
32. The surface area of the right square pyramid in the figure
below is given by S b
b
2
4
h
2
. If the pyramid has
height 10 feet and surface area 100 square feet, what is the
length of a side b of its base?
h
b
h
r
SECTION 2.3 Applications of Equations 113
33. A rope is to be stretched at uniform height from a tree to a
35-foot-long fence, which is 20 feet from the tree, and then
to the side of a building at a point 30 feet from the fence, as
shown in the figure. If 63 feet of rope is to be used, how far
from the building wall should the rope meet the fence?
34. Anne is standing on a straight road and wants to reach her
helicopter, which is located 2 miles down the road from her,
a mile from the road in a field (see the figure). She can run
5 miles per hour on the road and 3 miles per hour in the
field. She plans to run down the road, then cut diagonally
across the field to reach the helicopter. If she runs less than
a mile on the road and arrives at the helicopter in 42 minutes
(.7 hour), exactly how far did she run on the road?
35. A power plant is located on the bank of a river that is
1
2
mile wide. Wiring is to be laid across the river and then
along the shore to a substation 8 miles downstream, as
shown in the figure. It costs $12,000 per mile for underwa-
ter wiring and $8000 per mile for wiring on land. If $72,000
is to be spent on the project, how far from the substation
should the wiring come to shore?
Power plant
Substation
Anne
Helicopter
1 mile
Road
Field
30
20
Fence
Tree
36. Aspotlight is to be placed on the side of a 28-foot tall building
to illuminate a bench that is 32 feet from the base of the build-
ing. The intensity I of the light at the bench is known to be
x/d
3
, where x is the height of the spotlight above the ground
and d is the distance from the bench to the spotlight. If the
intensity is to be .00035, how high should the spotlight be?
37. In 2005 Dan Wheldon won the Indianapolis 500 (mile) race.
His speed was 83 mph faster than the speed of Ray Harroun,
who won the race in 1911. Wheldon took 3.53 hours less
than Harroun to complete the race. What was Wheldon’s
average speed?
38. A homemade loaf of bread turns out to be a perfect cube. Five
slices of bread, each .6 inch thick, are cut from one end of the
loaf. The remainder of the loaf now has a volume of 235
cubic inches. What were the dimensions of the original loaf?
39. A rectangular bin with an open top and volume of
38.72 cubic feet is to be built. The length of its base must be
twice the width, and the bin must be at least 3 feet high. Ma-
terial for the base of the bin costs $12 per square foot, and
material for the sides costs $8 per square foot. If it costs
$538.56 to build the bin, what are its dimensions?
114 CHAPTER 2 Graphs and Technology
THINKER
40. One corner of an 8.5 11 inch piece of paper is folded over
to the opposite side, as shown in the figure. The area of the
darkly shaded triangle at the lower left of the figure is 6
square inches and we want to find the length x.
(a) Take a piece of paper this size and experiment. Approx-
imately, what is the largest value x could have (and still
have the paper look like the figure)? With this value of
x, what is the approximate area of the triangle? Try
some other possibilities.
(b) Now find an exact answer by constructing and solving
a suitable equation. Explain why one of the solutions to
the equation is not an answer to this problem.
x
*This section may be omitted or postponed until Chapter 4 (Polynomial and Rational Functions). It
will be used only in occasional examples and in clearly identifiable exercises.
2.4 Optimization Applications*
■ Set up maximum and minimum problems.
■ Use graphing technology to find approximate solutions for such
problems.
Many real-life situations require you to find the largest or smallest quantity that
satisfies certain conditions. For instance, automotive engineers want to design
engines with maximum fuel efficiency. Similarly, a cereal manufacturer who
needs a box of volume 300 cubic inches might want to know the dimensions of the
box that requires the least amount of cardboard (and hence is cheapest). The exact
solutions of such minimum/maximum problems require calculus. However,
graphing technology can provide very accurate approximate solutions, as we shall
see. Before reading the following examples, you should review the guidelines for
setting up applied problems (page 101).
EXAMPLE 1
Sharon was at point A on the bank of a 2.5 kilometer wide river and traveled to
point B, 15 kilometers downstream on the opposite bank. She rowed at 4 kilo-
meters per hour to point C, and ran from C to B at 8 kilometers per hour. If
Section Objectives
she made the trip in the shortest possible time, what was that time and how far
did she run?
SOLUTION In Example 11 of Section 2.3, we showed that the total time y for
Sharon’s trip is given by
y
8
x
,
where x is the distance she ran. In the graph of this equation in Figure 2–51, each
point (x, y) on the graph represents one of the possibilities:
x represents the distance Sharon runs;
y represents her total time for the journey.
To find the shortest possible time, we must find the point on the graph with the
smallest y-coordinate, that is, the lowest point on the graph.
Therefore, the shortest time for the trip is 2.42 hours and occurs when Sharon runs
13.56 miles (that is, she lands 13.56 miles from B). ■
EXAMPLE 2
A landscaper wants to build a 10,000 square foot garden along the side of a river.
The rectangular garden will be fenced on three sides (no fence on the river side).
She plans to use ornamental steel fencing at $18 per foot on the side opposite the
river and chain link fence at $4 per foot on the other two sides. What dimensions
for the garden will minimize her cost? What is the minimum cost?
(x 1
5)
2
6.25
4
SECTION 2.4 Optimization Applications 115
15 − x
x
15
2.5
C
d
B
D
A
Figure 2–50
4
0
0
15
Figure 2–51
Graph the equation and use a minimum finder to verify that the lowest point is
approximately (13.56, 2.42).
GRAPHING EXPLORATION