Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach
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for some constant k. Since I 34.82, when d 2, we have
34.82
2
k
2
k 34.82(4) 139.28.
Therefore, the variation equation is
I
13
d
9.
2
28
.
So the light density at 10 feet is
I
13
1
9
0
.
2
28
1.3928 lumens per square foot. ■
In many situations, variation involves more than two variables. Typical
terminology in such cases is illustrated in the following table.
EXAMPLE 4
The electrical resistance R of wire (of uniform material) varies directly as the
length L and inversely as the square of the diameter d. A 2-meter-long piece
of wire with a diameter of 4.4 millimeters has a resistance of 500 ohms. What
diameter wire should be used if a 10-meter piece is to have a resistance of
1300 ohms?
SOLUTION The variation equation is
R
d
kL
2
for some constant k. So we substitute the known information (R 500 when
L 2 and d 4.4) and solve for k:
R
d
kL
2
500
(
k
4.
4)
2
2
k
500(
2
4.4)
2
4840.
36 CHAPTER 1 Basics
Example Terminology
I varies jointly as r and t.
I 100rt
I is jointly proportional to r and t.
H varies jointly as s and the cube of d.
H .06sd
3
H is jointly proportional to s and the cube of d.
P varies directly as T and inversely as V.
P
3
V
T
P is directly proportional to T and inversely
proportional to V.
S varies directly as w and the square of t and
inversely as ᐉ.
S
2.5
ᐉ
wt
2
S is directly proportional to w and the square
of t and inversely proportional to ᐉ.
Hence, the equation is
R
48
d
4
2
0L
.
We must find d when L 10 and R 1300:
1300
484
d
0
2
10
1300d
2
48,400
d
2
4
1
8
3
,4
0
0
0
0
37.2308.
Since the diameter is positive, we must have d 37.230
8
6.1 mm. ■
SPECIAL TOPICS 1.2.B Variation 37
EXERCISES 1.2.B
In Exercises 1–6, express each geometric formula as a state-
ment about variation by filling in the blanks in this sentence:
varies as ;
the constant of variation is .
1. Area of a circle of radius r: A pr
2
.
2. Volume of a sphere of radius r: V
4
3
pr
3
.
3. Area of a rectangle of length l and width w: A lw.
4. Area of a triangle of base b and height h: A
b
2
h
.
5. Volume of a right circular cone of base radius r and height
h: V
pr
3
2
h
.
6. Volume of a triangular cylinder of base b, height h, and
length l: V
1
2
bhl.
In Exercises 7–12, find the variation equation. Use k for the
constant of variation.
7. a varies inversely as b.
8. r is proportional to t.
9. z varies jointly as x, y, and w.
10. The weight w of an object varies inversely as the square of
the distance d from the object to the center of the earth.
11. The distance d one can see to the horizon varies directly as
the square root of the height h above sea level.
12. The pressure p exerted on the floor by a person’s shoe heel
is directly proportional to the weight w of the person and
inversely proportional to the square of the width r of the
heel.
b
h
l
In Exercises 13–22, express the given statement as an equation
and find the constant of variation.
13. v varies directly as u; v 8 when u 2.
14. v is directly proportional to u; v .4 when u .8.
15. v varies inversely as u; v 8 when u 2.
16. v is inversely proportional to u; v .12 when u .1.
17. t varies jointly as r and s; t 24 when r 2 and s 3.
18. B varies inversely as u and v; B 4 when u 1 and v 3.
19. w varies jointly as x and y
2
; w 96 when x 3 and y 4.
20. p varies directly as the square of z and inversely as r;
p 32/5 when z 4 and r 10.
21. T varies jointly as p and the cube of v and inversely as the
square of u; T 24 when p 3, v 2, and u 4.
22. D varies jointly as the square of r and the square of s and
inversely as the cube of t; D 18 when r 4, s 3, and t 2.
In Exercises 23–30, use an appropriate variation equation to
find the required quantity.
23. If r varies directly as t, and r 6 when t 3, find r when
t 2.
24. If r is directly proportional to t, and r 4 when t 2, find
t when r 2.
25. If b varies inversely as x, and b 9 when x 3, find b when
x 12.
26. If b is inversely proportional to x, and b 10 when x 4,
find x when b 12.
27. Suppose w is directly proportional to the sum of u and the
square of v. If w 200 when u 1 and v 7, then find
u when w 300 and v 5.
28. Suppose z varies jointly as x and y. If z 30 when x 5 and
y 2, then find x when z 45 and y 3.
29. Suppose r varies inversely as s and t. If r 12 when
s 3 and t 1, then find r when s 6 and t 2.
30. Suppose u varies jointly as r and s and inversely as t. If
u 1.5 when r 2, s 3, and t 4, then find r when
u 27, s 9, and t 5.
31. A resident of Michigan whose taxable income was $24,000
dollars paid state income tax of $936 in 2005.* Use the fact
that state income tax varies directly with the taxable income
to determine the state tax paid by someone with a taxable
income of $39,000.
32. By experiment, you discover that the amount of water that
comes from your garden hose varies directly with the water
pressure. A pressure of 10 pounds per square inch is needed
to produce a flow of 3 gallons per minute.
(a) What pressure is needed to produce a flow of 4.2 gallons
per minute?
(b) If the pressure is 5 pounds per square inch, what is the
flow rate?
33. According to Hooke’s law, the distance d that a spring
stretches when an object is attached to it is directly propor-
tional to the weight of the object. Suppose that the string
stretches 15.75 inches when a 7-pound weight is attached.
What weight is required to stretch the spring 27 inches?
34. In a thunderstorm, there is a gap between the time you see
the lightning and the time you hear the thunder. Your
distance from the storm is directly proportional to the time
interval between the lightning and the thunder. If you hear
thunder from a storm that is 1.36 miles away 6.6 seconds
after you see lightning, how far away is a storm for which
the time gap between lightning and thunder is 12 seconds?
35. At a fixed temperature, the pressure of an enclosed gas is
inversely proportional to its volume. The pressure is
50 kilograms per square centimeter when the volume
is 200 cubic centimeters. If the gas is compressed to
125 cubic centimeters, what is the pressure?
36. The electrical resistance in a piece of wire of a given length
and material varies inversely as the square of the diameter
of the wire. If a wire of diameter .01 cm has a resistance of
15.75 in.
7 lb
38 CHAPTER 1 Basics
.4 ohm, what is the resistance of a wire of the same length
and material but with diameter .025 cm?
37. The distance traveled by a falling object (subject only to
gravity, with wind resistance ignored) is directly propor-
tional to the square of the time it takes to fall that far. If an
object falls 100 feet in 2.5 seconds, how far does it fall in
5 seconds?
38. The weight of an object varies inversely with the square of
its distance from the center of the earth. Assume that the
radius of the earth is 3960 miles. If an astronaut weighs
180 pounds on the surface of the earth, what does that as-
tronaut weigh when traveling 300 miles above the surface
of the earth?
39. The weight of a cylindrical can of Glop varies jointly as the
height and the square of the base radius. The weight is
250 ounces when the height is 20 inches and the base
radius is 5 inches. What is the height when the weight is
960 ounces and the base radius is 8 inches?
40. The force of the wind blowing directly on a flat surface
varies directly with the area of the surface and the square of
the velocity of the wind. A 10 mile an hour (mph) wind
blowing on a wooden gate that measures 3 by 6 feet exerts
a force of 15 pounds. What is the force on a 1.5 by 2 foot
traffic sign from a 50 mph wind?
41. The force needed to keep a car from skidding on a circular
curve varies inversely as the radius of the curve and jointly
as the weight of the car and the square of the speed. It takes
1500 kilograms of force to keep a 1000-kilogram car from
skidding on a curve of radius 200 meters at a speed of
50 kilometers per hour. What force is needed to keep the
same car from skidding on a curve of radius 320 meters at
100 kilometers per hour?
42. Boyle’s law states that the volume of a given mass of gas
varies directly as the temperature and inversely as the pres-
sure. If a gas has a volume of 14 cubic inches when the tem-
perature is 40° and the pressure is 280 pounds per square
inch, what is the volume at a temperature of 50° and
pressure of 175 pounds per square inch?
43. The maximum safe load that a rectangular beam can sup-
port varies directly with the width w and the square of
the height h, and inversely with the length l of the beam.
A 6-foot-long beam that is 2 inches high and 4 inches wide
has a maximum safe load of 1000 pounds.
(a) What is the maximum safe load for a 10-foot-long beam
that is 4 inches high and 4 inches wide?
(b) How long should a 4 inch by 4 inch beam be to safely
support a 6000-pound load?
l
w
h
*The exercise assumes that there are no nonrefundable credits.
SECTION 1.3 The Coordinate Plane 39
Section Objectives
■ Locate points in the coordinate plane.
■ Create a scatter plot and line graph from a data set.
■ Determine the distance between two points in the plane.
■ Find the midpoint of a line segment.
■ Understand the relationship between equations and their graphs.
■ Find the intercepts of a graph.
■ Find the equation of a circle.
■ Identify equations whose graphs are circles.
Just as real numbers are identified with points on the number line, ordered pairs
of real numbers can be identified with points in the plane. To do this, draw two
number lines in the plane, one vertical and one horizontal, as in Figure 1–11. The
horizontal line is usually called the x-axis, and the vertical line the y-axis, but
other letters may be used if desired. The point where the axes intersect is the
origin. The axes divide the plane into four regions, called quadrants, that are
numbered as in Figure 1–11.
Figure 1–11
If P is a point in the plane, draw vertical and horizontal lines through P to the
coordinate axes, as shown in Figure 1–12, on the next page. These lines intersect the
x-axis at some number c and the y-axis at d. We say that P has coordinates (c, d ).
2
1
−11
Quadrant II Quadrant I
Quadrant III Quadrant IV
23456−2−3−4−5−6
−2
−3
−5
−6
−4
3
6
5
4
x
y
44. The period of a pendulum is the time it takes for the pendu-
lum to make one complete swing (going both left and right)
and return to its starting point. The period P varies directly
with the square root of the length l of the pendulum.
(a) Write the variation equation that describes this situa-
tion, using k for the constant of variation.
(b) What happens to the period if the length of the pen-
dulum is quadrupled?
1.3 The Coordinate Plane
The number c is the x-coordinate of P, and d is the y-coordinate of P. The plane
is said to have a rectangular (or Cartesian) coordinate system.
Figure 1–12
You can think of the coordinates of a point as directions for locating it. For
instance, to find (4, 3), start at the origin and move 4 units to the right along the
x-axis, then move 3 units downward, as shown in Figure 1–13, which also shows
other points and their coordinates.
Figure 1–13
EXAMPLE 1
The following table, from the U.S. Department of Education shows the maximum
Pell Grant for college students in selected years.
2
1
−11
(−5, 2)
(
0,
13
3
)
2345−2−3−4−5
−2
−3
−5
−4
3
5
4
x
Q
P
y
(
4
(2, 2)
(4, −3)
(2, −5)
(0, −3.5)
(−4, −1.2)
(−3, 0)
, 5
1
3
)
2
1
−1123456−2−3−4−5−6
−2
−3
−5
−6
−4
3
6
5
4
x
P
c
d
y
40 CHAPTER 1 Basics
Year 1990 1992 1994 1996 1998 2000 2002 2004
Amount 2300 2400 2300 2470 3000 3300 4000 4050
CAUTION
The coordinates of a point are an
ordered pair. Figure 1–13 shows that
the point P with coordinates (5, 2)
is quite different from the point Q with
coordinates (2, 5). The same
numbers (2 and 5) occur in both
cases, but in different order.
One way to represent this data graphically is to represent each year’s maximum
by a point; for instance (1990, 2300) and (2004, 4050). Alternatively, to avoid
using large numbers, we can let x be the number of years since 1990, so that
x 0 is 1990 and x 14 is 2004. We can also list the dollar amounts in hundreds,
so the points for 1990 and 2004 are (0, 23) and (14, 40.5). Plotting all the data in
this way leads to the scatter plot in Figure 1–14. Connecting these data points
with line segments produces the line graph in Figure 1–15. ■
SECTION 1.3 The Coordinate Plane 41
30
681024
20
10
40
x
y
12 14
Figure 1–14
30
681024
20
10
40
x
y
12 14
Figure 1–15
THE DISTANCE FORMULA
We shall often identify a point with its coordinates and refer, for example, to the
point (2, 3). When dealing with several points simultaneously, it is customary to
label the coordinates of the first point (x
1
, y
1
), the second point (x
2
, y
2
), the third
point (x
3
, y
3
), and so on.* Once the plane is coordinatized, it’s easy to compute the
distance between any two points:
Before proving the distance formula, we shall see how it is used.
EXAMPLE 2
To find the distance between the points (1, 3) and (2, 4) in Figure 1–16,
substitute (1, 3) for (x
1
, y
1
) and (2, 4) for (x
2
, y
2
) in the distance formula:
Distance formula: Distance
(x
1
x
2
)
2
(y
1
y
2
)
2
Substitute:
(1
2)
2
(3
(4))
2
Simplify:
(3)
2
(3
4)
2
9 1
10.
The Distance
Formula
The distance between points (x
1
, y
1
) and (x
2
, y
2
) is
(x
1
x
2
)
2
(y
1
y
2
)
2
.
y
−1
−4
−1−221
10
(−1, −3)
(2, −4)
x
Figure 1–16
*“x
1
” is read “x-one” or “x-sub-one”; it is a single symbol denoting the first coordinate of the first
point, just as c denotes the first coordinate of (c, d). Analogous remarks apply to y
1
, x
2
, and so on.
The order in which the points are used in the distance formula doesn’t make a dif-
ference. If we substitute (2, 4) for (x
1
, y
1
) and (1, 3) for (x
2
, y
2
), we get the
same answer:
[2 (
1)]
2
[4
(
3)]
2
3
2
(
1)
2
10. ■
EXAMPLE 3
In a Cubs game at Wrigley Field, a fielder catches the ball near the right-field
corner and throws it to second base. The right-field corner is 353 feet from home
plate along the right-field foul line. If the fielder is 5 feet from the outfield wall
and 5 feet from the foul line, how far must he throw the ball?
SOLUTION Imagine that the playing field is placed on the coordinate plane,
with home plate at the origin and the right-field foul line along the positive x-axis,
as shown in Figure 1–17 (not to scale).
Figure 1–17
Since the four bases form a square whose sides measure 90 ft each, second base
has coordinates (90, 90). The fielder is located 5 feet from the wall, so his
x-coordinate is 353 5 348. His y-coordinate is 5, since he is 5 feet from the
foul line. Therefore the distance he throws is the distance from (348, 5) to
(90, 90), which can be found as follows.
Distance formula: Distance
(x
1
x
2
)
2
(y
1
y
2
)
2
Substitute:
(348
90)
2
(5 9
0)
2
Simplify:
258
2
(85)
2
73,789
271.6 feet.
Therefore, he must throw about 272 feet. ■
Foul line
Wall
5
5
2nd base
1st base
3rd
base
Home
plate
90 353
355
90
42 CHAPTER 1 Basics
EXAMPLE 4
To find the distance from (a, b) to (2a, b), where a and b are fixed real numbers,
substitute a for x
1
, b for y
1
, 2a for x
2
, and b for y
2
in the distance formula:
(x
1
x
2
)
2
(y
1
y
2
)
2
(a 2
a)
2
(b (
b))
2
(a)
2
(b
b)
2
a
2
(
2b)
2
a
2
4
b
2
. ■
PROOF OF THE DISTANCE FORMULA
Figure 1–18 shows typical points P and Q in the plane. We must find length d of
line segment PQ.
Figure 1–18
As shown in Figure 1–18, the length of RQ is the same as the distance from
x
1
to x
2
on the x-axis (number line), namely, x
1
x
2
. Similarly, the length of PR
is the same as the distance from y
1
to y
2
on the y-axis, namely, y
1
y
2
. Accord-
ing to the Pythagorean Theorem* the length d of PQ is given by
(Length PQ)
2
(length RQ)
2
(length PR)
2
d
2
x
1
x
2
2
y
1
y
2
2
.
Since c
2
c
c c
2
c
2
(because c
2
0), this equation becomes
d
2
(x
1
x
2
)
2
(y
1
y
2
)
2
.
Since the length d is nonnegative, we must have
d
(x
1
x
2
)
2
(y
1
y
2
)
2
. ■
The distance formula can be used to prove the following useful fact (see
Exercise 96).
y
R
d
P(x
1
, y
1
)
Q(x
2
, y
2
)
⎜y
1
− y
2
⎜
y
2
y
1
⎜x
1
− x
2
⎜
x
1
x
2
x
SECTION 1.3 The Coordinate Plane 43
*See the Geometry Review Appendix.
The Midpoint
Formula
The midpoint of the line segment from (x
1
, y
1
) to (x
2
, y
2
) is
x
1
2
x
2
,
y
1
2
y
2
.
CAUTION
a
2
4
b
2
cannot be simplified. In
particular, it is not equal to a 2b.
EXAMPLE 5
To find the midpoint of the segment joining (1, 4) and (3, 1), use the formula in
the box with x
1
1, y
1
4, x
2
3, and y
2
1. The midpoint is
x
1
2
x
2
,
y
1
2
y
2
1
2
3
,
4
2
1
1,
5
2
as shown in Figure 1–19. ■
EXAMPLE 6
The annual revenues of the Dell Computer company were $31.2 billion in 2002
and $55.9 billion in 2006.* Assume that revenues are growing approximately lin-
early and estimate the revenues in 2004.
SOLUTION Let the point (x, y) denote the revenues y (in billions of dollars) in
year x. Then the points (2002, 31.2) and (2006, 55.9) represent the given data. The
midpoint of the line segment joining these points is
2002
2
2006
,
31.2
2
55.9
(2004, 43.55).
Since the data is growing linearly, this suggests that 2004 revenues were approx-
imately $43.55 billion. ■
GRAPHS
A graph is a set of points in the plane. Some graphs are based on data points, such
as Figures 1–14 and 1–15. Other graphs arise from equations, as follows. A
solution of an equation in variables x and y is a pair of numbers such that the sub-
stitution of the first number for x and the second for y produces a true statement.
For instance, (3, 2) is a solution of 5x 7y 1 because
5
3 7(2) 1,
and (2, 3) is not a solution because 5(2) 7
3 1. The graph of an equa-
tion in two variables is the set of points in the plane whose coordinates are solu-
tions of the equation. Thus the graph is a geometric picture of the solutions.
EXAMPLE 7
The graph of y x
2
2x 1 is shown in Figure 1–20. You can readily verify that
each of the points whose coordinates are labeled is a solution of the equation. For
instance, (0, 1) is a solution because 1 0
2
2(0) 1. ■
A graph may intersect the x- or y-axis at one or more points. The x-coordinate
of a point where the graph intersects the x-axis is called an x-intercept of the
graph. Similarly, the y-coordinate of a point where the graph intersects the y-axis
is called a y-intercept of the graph. Figure 1–21 shows some examples.
44 CHAPTER 1 Basics
y
2
−13
(3, 1)
(−1, 4)
x
(
1,
5
2
)
Figure 1–19
y
−2
1
2
3
4
−1−223
y = x
2
− 2x − 1
1
(0, −1)
(1.5, −1.75)
(−1, 2)
x
Figure 1–20
*Dell, Inc.
When the x- and y-intercepts cannot easily be read from the graph, they can
often be found algebraically.
EXAMPLE 8
Find the x- and y-intercepts of the graph of y x
2
2x 1 in Figure 1–20.
SOLUTION The points where the graph intersects the x-axis have 0 as their
y-coordinate (see Figure 1–20). We can find their x-coordinates by setting y 0
and solving the resulting equation,
x
2
2x 1 0.
By the quadratic formula
.4142
x
2
2
8
2.4142.
So the x-intercepts are approximately .4142 and 2.4142.
In this case, you can read the y-intercept from the graph; it is 1. Because
points on the y-axis have 0 as their x-coordinate, the y-intercept can be found
algebraically by setting x 0 in the equation and solving for y. ■
The process in Example 8 can be summarized as follows.
(2)
(2)
2
4
1
(
1)
2
1
SECTION 1.3 The Coordinate Plane 45
Figure 1–21
x-intercepts 3
y-intercepts 2
−2
2
4
−4
x
y
−4 −2−3 −1
2431
x-intercept 0
y-intercept 0
2
4
6
x
y
−4 −2−3 −1
2431
x-intercept 6
y-intercept 3
4
2
6
−2
x
y
−8 −4−6 −2
4862
Reading and interpreting information from graphs is an essential skill if you
want to succeed in this course and calculus.
EXAMPLE 9
Leslie Lahr takes out a 30-year mortgage on which her monthly payment is $850.
One of the graphs in Figure 1–22 shows the portion of each payment that goes to
interest and the other shows the portion that goes to paying off the principal.
x- and
y-Intercepts
To find the x-intercepts of the graph of an equation, set y 0
and solve for x.
To find the y-intercepts, set x 0 and solve for y.