Section 19–5
◆
531
Functions of More Than One Variable
Electrical
23. The current in a resistor is inversely proportional to the resistance. By what factor will the
current change if a resistor increases 10.0% due to heating?
24. The resistance of a wire is inversely proportional to the square of its diameter. If an AWG size
12 conductor (2.050-mm diameter) has a resistance of 14.8 , what will be the resistance of
an AWG size 10 conductor (2.588-mm diameter) of the same length and material?
25. The capacitive reactance XC of a circuit varies inversely as the capacitance C of the circuit. If the capacitance of a certain circuit is decreased by 25.0%, by what percentage will
XC change?
19–5 Functions of More Than One Variable
Joint Variation
So far in this chapter, we have considered only cases where y was a function of a single variable
x. In functional notation, this is represented by y f (x). In this section we cover functions of
two or more variables, such as
y f (x, w)
y f (x, w, z)
and so forth. When y varies directly as x and w, we say that y varies jointly as x and w. The
three variables are related by the following equation, where, as before, k is a constant of
proportionality:
Joint
Variation
y = kxw
or
y xw
62
Example 27: If y varies jointly as x and w, how will y change when x is doubled and w is
one-fourth of its original value?
◆◆◆
Solution: Let y be the new value of y obtained when x is replaced by 2x and w is replaced by
w4, while the constant of proportionality k, of course, does not change. Substituting in Eq. 62,
we obtain
w
y k (2x) p q
4
kxw
2
But, since kxw y, then
y
y 2
So the new y is half as large as its former value.
◆◆◆
Combined Variation
When one variable varies with two or more variables in ways that are more complex than in
Eq. 62, it is referred to as combined variation. This term is applied to many different kinds of
relationships, so a formula cannot be given that will cover all types. You must carefully read the
problem statement in order to write the equation. Once you have the equation, the solution of
combined variation problems is no different than for other types of variation.
532
Chapter 19
◆◆◆
◆
Ratio, Proportion, and Variation
Example 28: If y varies directly as the cube of x and inversely as the square of z, we write
kx3
y z2
◆◆◆
Example 29: If y is directly proportional to x and the square root of w, and inversely proportional to the square of z, we write
◆◆◆
kxw
y z2
◆◆◆
Example 30: If y varies directly as the square root of x and inversely as the square of w, by
what factor will y change when x is made four times larger and w is tripled?
◆◆◆
Solution: First write the equation linking y to x and w, including a constant of proportionality k.
k x
y w2
We get a new value for y (let’s call it y) when x is replaced by 4x and w is replaced by 3w. Thus
2x
k 4x
y k 2
9w2
(3w)
2 p k x q 2
y
9 w2
9
2
We see that y is 9 as large as the original y.
◆◆◆
Example 31: If y varies directly as the square of w and inversely as the cube root of x, and
y 225 when w 103 and x 157, find y when w 126 and x 212.
◆◆◆
Solution: From the problem statement,
w2
y k 3 x
Solving for k, we have
(103)2
225 k 1967k
3 157
or k 0.1144. So
0.1144w2
y 3 x
When w 126 and x 212,
0.1144(126)2
y 305
3 212
◆◆◆
Example 32: y is directly proportional to the square of x and inversely proportional to the
cube of w. By what factor will y change if x is increased by 15% and w is decreased by 20%?
◆◆◆
Solution: The relationship between x, y, and w is
x2
y k w3
so we can write the proportion
x22
k y2
w23
x22 w12 p x2 q2 p w1 q3
y1
x12
w23 x13
x1
w2
k 3
w1
Section 19–5
◆
533
Functions of More Than One Variable
We now replace x2 with 1.15x1 and replace w2 with 0.8w1.
y2
y1
p 1.15x1 q p
2
x1
w1
0.8w1
q
3
(1.15)2
2.58
(0.8)3
So y will increase by a factor of 2.58.
Exercise 5
◆
◆◆◆
Functions of More Than One Variable
Joint Variation
1. If y varies jointly as w and x, and y is 483 when x is 742 and w is 383, find y when x is 274
and w is 756.
2. If y varies jointly as x and w, by what factor will y change if x is tripled and w is halved?
3. If y varies jointly as w and x, by what percent will y change if w is increased by 12% and x
is decreased by 7.0%?
4. If y varies jointly as w and x, and y is 3.85 when w is 8.36 and x is 11.6, evaluate the constant
of proportionality, and write the complete expression for y in terms of w and x.
5. If y varies jointly as w and x, fill in the missing values.
w
x
y
46.2
18.3
127
19.5
41.2
8.86
155
12.2
79.8
Combined Variation
6. If y is directly proportional to the square of x and inversely proportional to the cube of w,
and y is 11.6 when x is 84.2 and w is 28.4, find y when x is 5.38 and w is 2.28.
7. If y varies directly as the square root of w and inversely as the cube of x, by what factor will
y change if w is tripled and x is halved?
8. If y is directly proportional to the cube root of x and to the square root of w, by what percent
will y change if x and w are both increased by 7.0%?
9. If y is directly proportional to the 32 power of x and inversely proportional to w, and y is 284
when x is 858 and w is 361, evaluate the constant of proportionality, and write the complete
equation for y in terms of x and w.
10. If y varies directly as the cube of x and inversely as the square root of w, fill in the missing
values in the table.
w
x
1.27
y
3.05
5.66
1.93
4.66
2.75
3.87
7.07
1.56
534
Chapter 19
◆
Ratio, Proportion, and Variation
Geometry
11. The area of a triangle varies jointly as its base and altitude. By what percent will
the area change if the base is increased by 15% and the altitude decreased by 25%?
12. If the base and the altitude of a triangle are both halved, by what factor will the area
change?
Electrical
13. When an electric current flows through a wire, the resistance to the flow varies directly
as the length and inversely as the cross-sectional area of the wire. If the length and the
diameter are both tripled, by what factor will the resistance change?
_
14. If 750 m of 3.00-mm-diameter wire has a resistance of 27.6 , what length of similar wire
5.00 mm in diameter will have the same resistance?
Gravitation
15. Newton’s law of gravitation states that every body in the universe attracts every other body
with a force that varies directly as the product of their masses and inversely as the square
of the distance between them. By what factor will the force change when the distance is
doubled and each mass is tripled?
16. If both masses are increased by 60% and the distance between them is halved, by what
percent will the force of attraction increase?
Illumination
17. The intensity of illumination at a given point is directly proportional to the intensity of the
light source and inversely proportional to the square of the distance from the light source.
If a desk is properly illuminated by a 75.0-W lamp 2.50 m from the desk, what size lamp
will be needed to provide the same lighting at a distance of 3.75 m?
18. How far from a 150-candela light source would a picture have to be placed so as to receive
the same illumination as when it is placed 12 m from an 85-candela source?
Gas Laws
19. The volume of a given weight of gas varies directly as the absolute temperature t and
inversely as the pressure p. If the volume is 4.45 m3 when p 225 kilopascals (kPa) and
t 305 K, find the volume when p 325 kPa and t 354 K.
20. If the volume of a gas is 125 m3, find its volume when the absolute temperature is increased
10% and the pressure is doubled.
Work
21. The amount paid to a work crew varies jointly as the number of persons working and the
length of time worked. If 5 workers earn $5,123.73 in 3.0 weeks, in how many weeks will
6 workers earn a total of $6,148.48?
22. If 5 bricklayers take 6.0 days to finish a certain job, how long would it take 7 bricklayers
to finish a similar job requiring 4 times the number of bricks?
Strength of Materials
23. The maximum safe load of a rectangular beam (Fig. 19–12) varies jointly as the width
and the square of the depth and inversely as the length of the beam. If a beam 8.00 cm
Section 19–5
◆
535
Functions of More Than One Variable
_
wide, 11.5 cm deep, and 2.00 m long can safely support 70 00 kg, find the safe load
for a beam 6.50 cm wide, 13.4 cm deep, and 2.60 m long made of the same material.
Length
Depth
Width
FIGURE 19–12 A rectangular beam.
24. If the width of a rectangular beam is increased by 11%, the depth decreased by 8%, and the
length increased 6%, by what percent will the safe load change?
Mechanics
25. The number of vibrations per second made when a stretched wire (Fig. 19–13) is plucked
varies directly as the square root of the tension in the wire and inversely as the length. If
a 1.00-m-long wire will vibrate 325 times a second when the tension is 115 N, find the
frequency of vibration if the wire is shortened to 0.750 m and the tension is decreased to
95.0 N.
Tension
FIGURE 19–13 A stretched wire.
26. The kinetic energy of a moving body is directly proportional to its mass and the square of
its speed. If the mass of a bullet is halved, by what factor must its speed be increased to
have the same kinetic energy as before?
Fluid Flow
27. The time needed to empty a vertical cylindrical tank (Fig. 19–14) varies directly as the
square root of the height of the tank and the square of the radius. By what factor will the
emptying time change if the height is doubled and the radius increased by 25%?
28. The power available in a jet of liquid is directly proportional to the cross-sectional area of
the jet and to the cube of the velocity. By what factor will the power increase if the area
and the velocity are both increased 50%?
Case Study Discussion—Wheatstone Bridge
The strain gauge has a nominal resistance of 1000 . When stressed, the resistance will
change. Now we could just measure the current flow through the gauge, and divide that
into our applied voltage (by Ohm’s law) to give us the resistance. The problem is that
the accuracy of the measurement depends not only on a very good ammeter (to measure
current) but also on a very precise voltage source. The Wheatstone bridge operates on the
principle of ratios of the resistances that form the bridge. Regardless of the power supply,
we simply measure VOUT and VIN and use ratios. Even if the 9V in our example varied, the
VOUT would also vary but the ratio would be the same.
FIGURE 19–14 A vertical
cylindrical tank.
536
Chapter 19
◆
Ratio, Proportion, and Variation
R4
R2 q
p
VOUT VIN
R3 R4 R1 R2
R4
1000
p
q
0.012 9
1000 R4 1000 1000
0.012
9
p
R4
1000 q
2000
1000 R4
•
•
•
501.33 R4 0.501 33 R4
•
•
501.33 R4 (1 0.501 33)
R4
•
0.001 33 0.5
1000 R4
501.33 R4(0.498 66)
R4
•
0.501 33 1000 R4
• R4
•
0.501 33(1000 R4) R4
◆◆◆
A knot is equal to 1 nautical mile
(M) per hour (1 M 1852 m).
•
501.33 0.501 33 R4 R4
•
•
•
501.33
0.498 66
1005.347 R4
CHAPTER 19 REVIEW PROBLEMS ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆
1. If y varies inversely as x, and y is 736 when x is 822, find y when x is 583.
2. If y is directly proportional to the 52 power of x, by what factor will y change when x is
tripled?
3. If y varies jointly as x and z, by what percent will y change when x is increased by 15% and
z is decreased by 4%?
4. The braking distance of an automobile varies directly as the square of the speed. If the
braking distance of a certain automobile is 11.0 m at 40.0 kmh, find the braking distance
at 90.0 kmh.
5. The rate of flow of liquid from a hole in the bottom of a tank is directly proportional to the
square root of the liquid depth. If the flow rate is 225 Lmin when the depth is 3.46 m, find
the flow rate when the depth is 1.00 m.
6. The power needed to drive a ship varies directly as the cube of the speed of the ship, and
a 77.4-kW engine will drive a certain ship at 11.2 knots (kn). Find the power needed to
propel that ship at 18.0 kn.
7. If the tensile strength of a cylindrical steel bar varies as the square of its diameter, by what
factor must the diameter be increased to triple the strength of the bar?
8. The life of an incandescent lamp varies inversely as the 12th power of the applied voltage, and the light output varies directly as the 3.5th power of the applied voltage. By what
factor will the life increase if the voltage is lowered by an amount that will decrease the
light output by 10%?
9. One of Kepler’s laws states that the time for a planet to orbit the sun varies directly as the 32
power of its distance from the sun. How many years will it take for Saturn, which is about
912 times as far from the sun as is the earth, to orbit the sun?
10. The volume of a cone varies directly as the altitude and the square of the base radius.
By what factor will the volume change if the altitude is doubled and the base radius is
halved?
11. The number of oscillations made by a pendulum in a given time is inversely proportional
to the length of the pendulum. A certain clock with a 75.00-cm-long pendulum is losing 15.00 mind. Should the pendulum be lengthened or shortened, and by how much?
12. A trucker usually makes a trip in 18.0 h at an average speed of 90.0 kmh. Find the travelling time if the speed were reduced to 75.0 kmh.