2.6
Variation
2.6
OBJECTIVES
1. Solve problems involving direct variation
2. Solve problems involving inverse variation
3. Solve problems involving direct, inverse, and
joint variation
NOTE Considering these
special functions is called the
study of variation because the
quantity varies.
NOTE The idea that the
circumference is a constant
times the diameter was known
to the Babylonians as early as
2000 B.C.
We have seen that equations can describe a relationship between two quantities. One particular type of equation is so common that there exists a precise terminology to describe the
relationship. That type of equation is our focus in this section. In each case we will see that
a quantity is varying with respect to one or more other quantities in a particular fashion.
Suppose that our quantities are related in a manner such that one is a constant multiple
of the other. There are many real-world applications.
The circumference of a circle is a constant multiple of the length of its diameter:
C pd
Circumference
Diameter
The constant pi
If a rate or speed is constant, the distance traveled is a constant multiple of time.
d rt
Distance
Time
Rate—a constant
If you earn a fixed hourly pay, your total pay is a constant multiple of the number of hours
that you worked.
T ph
Total pay
Hours worked
Hourly pay—a constant
In all the cases above, the changes in one variable are proportional to the changes in the
other. For instance, if the diameter of a circle is doubled, its circumference is doubled. This
leads to our first definition.
Definitions: Direct Variation
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If y is a constant multiple of x, we write
y kx
k is a constant
and say that y varies directly as x, or that y is directly proportional to x. The
constant k is called the constant of variation.
Typically, in a variation problem, you will be given the type of variation involved and related values for the variables. From this information you can determine the constant of variation and, therefore, the equation relating the quantities. The following examples illustrate.
101
102
CHAPTER 2
LINEAR EQUATIONS AND INEQUALITIES
Example 1
Solving a Direct Variation Problem
If y varies directly as x and y 40 when x 8, find the equation relating x and y. Also find
the value of y when x 10.
Because y varies directly with x, from our definition we have
NOTE As you will see, in direct
variation, as the absolute value
of one variable increases, the
absolute value of the other also
increases.
y kx
We need to determine k, and this is easily done by letting x 8 and y 40:
40 k(8)
k5
The desired equation relating x and y is then
y 5x
To complete the example, if x 10,
y 5(10) 50
CHECK YOURSELF 1
If y varies directly as x and y 72 when x 9, find the value of y when x 12.
As we said, problems of variation occur frequently in applications of mathematics to
many fields. The following is a typical example.
Example 2
Solving a Direct Variation Application
application of Hooke’s law.
In physics, it is known that the force F needed to stretch a spring x units varies directly with
x. If a force of 18 lb stretches a spring 3 in., how far will the same spring be stretched by a
force of 30 lb?
From the problem we know that F kx, and letting F 18 and x 3, we have
18 k(3)
k6
Therefore, F 6x relates the two variables. Now, to find x when F 30, we write
30 6x
x5
So the force of 30 lb would stretch the spring 5 in.
CHECK YOURSELF 2
The pressure at a point under water is directly proportional to the depth. If the
pressure at a depth of 2 ft is 125 lb/ft2, find the pressure at a depth of 10 ft.
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NOTE Example 2 is an
VARIATION
SECTION 2.6
103
Many applications require that one variable be directly proportional to some power of a
second variable. For instance, we might say that y is directly proportional to the square of x
and write
NOTE We also say that “y
y kx2
varies directly with the square
of x.”
Consider the following example.
Example 3
Solving a Direct Variation Problem
The distance s that an object will fall from rest (neglecting air resistance) varies directly
with the square of the time t. If an object falls 64 ft in 2 s, how far will it fall in 5 s?
The relating equation in this example is
s kt2
By letting s 64 and t 2, we can determine k:
64 k(2)2
64 4k
so
k 16
We now know that the desired equation is s 16t 2, and substituting 5 for t, we have
s 16(5)2 400 ft
CHECK YOURSELF 3
The distance that an object falls from rest varies directly with the square of time t.
If an object falls 144 ft in 3 s, how far does it fall in 6 s?
NOTE Perhaps the most
common example is the
relationship between rate and
time. The faster something
travels (rate increases), the
sooner it arrives (time
decreases).
If two quantities are related so that an increase in the absolute value of the first gives a
proportional decrease in the absolute value of the second, we say that the variables vary inversely with each other. This leads to our second definition.
Definitions: Inverse Variation
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If y varies inversely as x, we write
y
k
x
k is a constant
We can also say that y is inversely proportional to x.
Example 4
Solving an Inverse Variation Problem
1
If y varies inversely as x and y 18 when x , find the equation relating x and y. Also
2
find the value of y when x 3.
CHAPTER 2
LINEAR EQUATIONS AND INEQUALITIES
Because y varies inversely as x, we can write
y
k
x
1
Now with y 18 and x , we have
2
18 k
1
2
9k
We now have the desired equation relating x and y:
y
9
x
and when x 3,
y
9
3
3
CHECK YOURSELF 4
If w is inversely proportional to z and w 15 when z 5, find the equation relating
z and w. Also find the value of w when z 3.
Let’s consider an application that involves the idea of inverse variation.
Example 5
Solving an Inverse Variation Application
The intensity of illumination I of a light source varies inversely as the square of the distance
d from that source. If the illumination 4 ft from the source is 9 footcandles (fc), find the
illumination 6 ft from the source.
4 ft
6 ft
From the given information we know that
I
k
d2
Letting d 4 and I 9, we first find the constant of variation k:
9
k
42
and
k 144
or
9
k
16
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104
VARIATION
SECTION 2.6
105
The equation relating I and d is then
I
144
d2
and when d 6,
I
144
144
4 fc
2 6
36
CHECK YOURSELF 5
At a constant temperature, the volume of a gas varies inversely as the pressure. If
a gas has volume 200 ft3 under a pressure of 40 pounds per square inch (lb/in.2),
what will be its volume under a pressure of 50 lb/in.2?
It is also common for one quantity to depend on several others. We can find a familiar
example from geometry.
The volume of a cylinder depends on its height and the square of its radius.
This is an example of joint variation. We say that the volume V varies jointly with the
height h and the square of the radius r. We can write
NOTE You probably recognize
V khr2
that k, the constant of
variation, is p in this case.
In general:
Definitions: Joint Variation
If z varies jointly as x and y, we write
z kxy
k is a constant
The solution techniques for problems involving joint variation are similar to those used
earlier, as Example 6 illustrates.
Example 6
Solving a Joint Variation Problem
Assume that z varies jointly as x and y. If z 100 when x 2 and y 20, find the value of
z if x 4 and y 30.
From the given information we have
z kxy
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Letting z 100, x 2, and y 20 gives
100 k(2)(20)
or
k
5
2
The equation relating z with x and y is then
5
z xy
2
and when x 4 and y 30, by substitution
z
5
(4)(30) 300
2
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CHAPTER 2
LINEAR EQUATIONS AND INEQUALITIES
CHECK YOURSELF 6
Assume that r varies jointly as s and t. If r 64 when s 3 and t 8, find the value
of r when s 16 and t 12.
Once again there are many physical applications of the concept of joint variation. The
following is a typical example.
Example 7
Solving a Joint Variation Application
The “safe load” for a wooden rectangular beam varies jointly as its width and the square of
its depth.
If the safe load of a beam 2 in. wide and 8 in. deep is 640 lb, what is the safe load of a
beam 4 in. wide and 6 in. deep?
2 in.
8 in.
From the given information,
S kwd 2
Substituting the given values yields
640 k(2)(8)2
k5
We then have the equation
S 5wd 2
and, for the 4 by 6 in. beam,
S 5(4)(6)2 720 lb
CHECK YOURSELF 7
The force of a wind F blowing on a vertical wall varies jointly as the surface area of
the wall A and the square of the wind velocity v.
If a wind of 20 mi/h has a force of 100 lb on a wall with area 50 ft2, what force
will a wind of 40 mi/h produce on the same wall?
There is one final category of variation problems. This category involves applications
in which inverse variation is combined with direct or joint variation in stating the equation
relating the variables.
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NOTE S is the safe load, w the
width, and d the depth.
VARIATION
SECTION 2.6
107
These are called combined variation problems. In general, a typical statement form is as
follows:
Definitions: Combined Variation
If z varies directly as x and inversely as y, we write
z
kx
y
in which k is a constant
Example 8
Solving a Combined Variation Problem
Assume that w varies directly as x and inversely as the square of y. When x 8 and y 4,
w 18. Find w if x 4 and y 6.
From the given information we can write
w
kx
y2
Substituting the known values, we have
18 k8
42
or
k 36
We now have the equation
w
36x
y2
and letting x 4 and y 6, we get
w
36 4
4
62
CHECK YOURSELF 8
Ohm’s law for an electric circuit states that the current I varies directly as the
electromotive force E and inversely as the resistance R.
If the current is 10 amperes (A), the electromotive force is 110 volts (V) and the
resistance is 11 ohms (
). Find the current for an electromotive force of 220 V and a
resistance of 5 .
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Translating verbal problems to algebraic equations is the basis for all variation applications. The following table gives some typical examples from our work in this section.
Variation Statement
Algebraic Equation
y varies directly as x.
y kx
k
y
x
z kxy
kx
z
y
y varies inversely as x.
z varies jointly as x and as y.
z varies directly as x and inversely as y.
CHAPTER 2
LINEAR EQUATIONS AND INEQUALITIES
All four basic types of variation problems involve essentially the same solution technique. The following algorithm summarizes the steps involved in all the variation problems
that we have considered.
Step by Step: Solving Problems Involving Variation
Step 1 Translate the given problem to an algebraic equation involving the
constant of variation k.
Step 2 Use the given values to find that constant.
Step 3 Replace k with the value found in step 2 to form the general equation
relating the variables.
Step 4 Substitute the appropriate values of the variables to solve for the
corresponding value of the desired unknown quantity.
CHECK YOURSELF ANSWERS
1. y 96
2. 625 lb/ft2
5. 160 ft3
6. 512
3. 576 ft
7. 400 lb
4. w 8. I 75
, w 25
z
kE
and k 1, so I 44 A
R
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108
Name
2.6
Exercises
Section
Date
Translate each of the following statements of variation to an algebraic equation, using k
as the constant of variation.
ANSWERS
1. s varies directly as the square of x.
1.
2. z is directly proportional to the square root of w.
2.
3.
3. r is inversely proportional to s.
4.
4. m varies inversely as the cube of n.
5.
6.
5. V varies directly as T and inversely as P.
7.
8.
6. A varies jointly as x and y.
9.
7. V varies jointly as h and the square of r.
10.
11.
8. t is directly proportional to d and inversely proportional to r.
12.
9. w varies jointly as x and y and inversely as the square of z.
13.
14.
10. p varies jointly as r and the square of s and inversely as the cube of t.
Find k, the constant of variation, given each of the following sets of conditions.
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11. y varies directly with x; y 54 when x 6.
12. m varies inversely with p; m 5 when p 3.
13. r is inversely proportional to the square of s; r 5 when s 4.
14. u varies directly with the square of w; u 75 when w 5.
109
ANSWERS
15.
15. V varies jointly as x and y; V 100 when x 5 and y 4.
16.
16. w is directly proportional to u and inversely proportional to v; w 20 when u 10
17.
and v 3.
18.
19.
17. z varies directly as the square of x and inversely as y; z 20 when x 2 and y 4.
20.
18. p varies jointly as r and the square of q; p 144 when q 6 and r 2.
21.
22.
19. m varies jointly as n and the square of p and inversely as r; m 40 when n 5,
p 2, and r 4.
23.
24.
20. x varies directly as the square of y and inversely as w and z; x 8 when y 4,
w 3, and z 2.
25.
26.
Solve each of the following variation problems.
27.
21. Let y vary directly with x. If y 60 when x 5, find the value of y when x 8.
22. Suppose that z varies inversely as the square of w and that z 3 when w 4. Find
the value of z when w 6.
23. Variable A varies jointly with x and y, and A 120 when x 6 and y 5. Find the
value of A when x 8 and y 3.
24. Let p be directly proportional to q and inversely proportional to the square of r. If
p 3 when q 8 and r 4, find p when q 9 and r 6.
25. Suppose that s varies directly with r and inversely with the square of t. If s 4 when
26. Variable p varies jointly with the square root of r and the square of q. If p 72 when
r 16 and q 3, find the value of p when r 25 and q 2.
Solve each of the following variation problems.
27. The length that a spring will stretch varies directly as the force applied to the spring.
If a force of 10 lb will stretch a spring 2 in., what force will stretch the same spring
3 in.?
110
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r 12 and t 6, find the value of s when r 8 and t 4.
ANSWERS
28. If the temperature of a gas is held constant, the volume occupied by that gas varies
inversely as the pressure to which the gas is subjected. If the volume of a gas is
8 ft3 when the pressure is 12 lb/in.2, find the volume of the gas if the pressure is
16 lb/in.2.
28.
29.
30.
31.
29. If the current, in amperes, in an electric circuit is inversely proportional to the
resistance, the current is 55 A when the resistance is 2 . Find the current when the
resistance is 5 .
32.
33.
30. The distance that a ball rolls down an inclined plane varies directly as the
34.
square of the time. If the ball rolls 36 ft in 3 s, how far will it roll in 5 s?
35.
31. The volume of a right circular cone varies jointly as the height and the square
36.
of the radius. If the volume of the cone is 15p cm3 when the height is 5 cm
and the radius is 3 cm, find the volume when the height is 6 cm and the radius
is 4 cm.
37.
32. The safe load of a rectangular beam varies jointly as its width and the square
of its depth. If the safe load of a beam is 1000 lb when the width is 2 in. and
the depth is 10 in., find its safe load when the width is 4 in. and the depth is
8 in.
33. The stopping distance (in feet) of an automobile varies directly as the square
of its speed (in miles per hour). If a car can stop in a distance of 80 ft from
40 mi/h, how much distance will it take to stop from a speed of 60 mi/h?
34. The period (the time required for one complete swing) of a simple pendulum
is directly proportional to the square root of its length. If a pendulum with
length 9 cm has a period of 3.3 s, find the period of a pendulum with length
16 cm.
35. The distance (in miles) that a person can see to the horizon from a point above
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Earth’s surface is directly proportional to the square root of the height (in feet)
of that point. If a person 100 ft above Earth’s surface can see 12.5 mi, how far
can an observer in a light airplane at 3600 ft see to the horizon?
36. The illumination produced by a light source on a surface varies inversely as
the square of the distance of that surface from the source. If a light source
produces an illumination of 48 fc on a wall 4 ft from the source, what will be
the illumination (in footcandles) of a wall 8 ft from the source?
37. The electrical resistance of a wire varies directly as its length and inversely as
the square of its diameter. If a wire with length 200 ft and diameter 0.1 in. has
a resistance of 2 , what will be the resistance in a wire of length 400 ft with
diameter 0.2 in.?
111
ANSWERS
38.
38. The frequency of a guitar string varies directly as the square root of the
tension on the string and inversely as the length of the string. If a frequency of
440 cycles per second, or hertz (Hz), is produced by a tension of 36 lb on a
string of length 60 cm, what frequency (in hertz) will be produced by a tension
of 64 lb on a string of length 40 cm?
39.
40.
39. The temperature of the steam from a geothermal source is inversely
proportional to the distance it is transported. Write an algebraic equation
relating temperature to distance.
40. Power available from a wind generator varies jointly as the square of the
diameter of the rotor and the cube of the wind velocity. Write an algebraic
equation relating power, rotor diameter, and wind speed.
Answers
kT
kxy
7. V khr 2
9. w 2
P
z
11. 9
13. 80
15. 5
17. 20
19. 8
21. 96
23. 96
25. 6
27. 15 lb
29. 22 A
31. 32p cm3
33. 180 ft
k
35. 75 mi
37. 1 39. t d
3. r k
s
5. V © 2001 McGraw-Hill Companies
1. s kx2
112