526
Chapter 19
◆
Ratio, Proportion, and Variation
34. If it cost $756 to put a fence around a circular pond, what will it cost to enclose another
pond having 15 the area?
35. A triangular field whose base is 215 m contains 12 400 m2. Find the area of a field of similar form whose base is 328 m.
Fluids
36. At the Darlington nuclear plant in Ontario, operators must know the rate of flow in the
pipes cooling the reactors. However, for operational and safety reasons, the pipes cannot
be blocked by a measuring instrument. Instead, the pressure is measured before and after a
narrowing of the pipe (see Fig. 19–9). The rate of flow is directly proportional to the square
root of the difference in pressure P, where P P2 P1
(a) Write the expression for rate of flow.
(b) The flow in a pipe is 200 L /min. If P increases by 16%, what is the new flow rate?
P1
P2
FIGURE 19–9 Diaphragms measure
pressure without blocking flow.
19–4
Inverse Variation
Definition
When we say that “y varies inversely as x” or that “y is inversely proportional to x,” we
mean that x and y are related by the following equation, where, as before, k is a constant of
proportionality:
We could also say that we have
inverse variation when the
product of x and y is a constant,
that is, when xy k.
Inverse
Variation
k
y
x
or
1
y
x
61
The Hyperbola
The graph of the function y kx is called a hyperbola. It is one of the conic sections, which
you will study in more detail when you study analytic geometry (Chapter 22).
◆◆◆
Example 21: Graph the function y 2x from x 0 to x 5.
Solution: Make a table of point pairs.
Do not confuse inverse variation
with the inverse of a function.
x
0
1
2
1
2
3
4
5
y
Not
defined
4
2
1
2
3
1
2
2
5
These points are graphed in Fig. 19–10. Note that as x takes on smaller and smaller values, y
gets larger and larger; and as x gets very large, y gets smaller and smaller but never reaches zero.
Section 19–4
◆
527
Inverse Variation
Thus the curve gets ever closer to the x and y axes but never touches them. The x and y axes are
asymptotes for this curve.
y
6
5
4
y=2
x
3
2
1
−5
−4
−3
−2
−1
0
1
2
3
4
5 x
−1
−2
−3
−4
FIGURE 19–10
A hyperbola.
We did not plot the curve for negative values of x, but, if we had, we would have gotten the
second branch of the hyperbola, shown dashed in the figure.
Power Function with Negative Exponent
Suppose, for example, that y varies inversely as the cube of x. This would be written
k
y
x3
which is the same as
y kx3
Note that this equation is a power function with a negative exponent. Such power functions are
sometimes referred to as hyperbolic.
◆◆◆
Example 22: Rewrite yx2 10.
Solution: Dividing by x2, we see that
10
y x2
which can be rewritten as
y 10x2
Therefore y is 10 times the inverse proportion of the square of x.
◆◆◆
Solving Inverse Variation Problems
Inverse variation problems are solved by the same methods as for any other power function.
As before, we can work these problems with or without finding the constant of proportionality.
528
Chapter 19
◆◆◆
◆
Ratio, Proportion, and Variation
Example 23: If y is inversely proportional to x, and y is 8 when x is 3, find y when x is 12.
Solution: x and y are related by y kx. We find the constant of proportionality by substituting
the given values of x and y (3 and 8).
k
8
3
k 3(8) 24
So y 24x. When x 12,
24
y 2
12
◆◆◆
◆◆◆ Example 24: If y is inversely proportional to the square of x, by what factor will y change
if x is tripled?
Solution: From the problem statement we know that y kx2 or k x2y. If we use the subscript
1 for the original values and 2 for the new values, we can write
k x21y1 x22 y2
Since the new x is triple the old, we substitute 3x1 for x2.
x21 y1 (3x1)2y2
9x21y2
y1 9y2
or
y1
y2 9
So the new y is one-ninth of its initial value.
◆◆◆
◆◆◆
Example 25: If y varies inversely as x, fill in the missing values in the table:
x
1
y
9
12
12
4
Solution: Substitute the ordered pair (9, 4) into Eq. 61.
k
4
9
k 36
So
36
y x
Filling in the table, we have
x
1
3
9
12
y
36
12
4
3
Section 19–4
◆
529
Inverse Variation
◆◆◆ Example 26: The number of oscillations N that a pendulum (Fig. 19–11) makes per unit
time is inversely proportional to the square root of the length L of the pendulum, and N 2.00
oscillations per second when L 85.0 cm.
(a) Write the equation N f (L).
(b) Find N when L 115 cm.
L
Solution:
(a) The equation will be of the form
k
N L
FIGURE 19–11 A simple
pendulum.
Substituting N 2.00 and L 85.0 gives us
k
2.00 85.0
k 2.00 85.0 18.4
So the equation is
18.4
N L
(b) When L 115,
18.4
N 1.72 oscillations per second
115
Check: Does the answer seem reasonable? We see that N has decreased, which we expect with
a longer pendulum, but it has not decreased by much. As a check, we note that L has increased
by a factor of 115 85, or 1.35, so we expect N to decrease by a factor of 1.35 , or 1.16.
◆◆◆
Dividing, we find that 2.00 1.72 1.16, so the answer checks.
Exercise 4
◆
Inverse Variation
1. If y varies inversely as x, and y is 385 when x is 832, find y when x is 226.
2. If y is inversely proportional to the square of x, and y has the value 1.55 when x is 7.38, find
y when x is 44.2.
3. If y is inversely proportional to x, how does y change when x is doubled?
4. If y varies inversely as x, and y has the value 104 when x is 532:
(a) Find the constant of proportionality.
(b) Write the equation y f (x).
(c) Find y when x is 668.
(d) Find x when y is 226.
5. If y is inversely proportional to x, fill in the missing values.
x
306
y
125
622
418
6. Fill in the missing values, assuming that y is inversely proportional to the square of x.
x
y
2.69
7.83
1.16
1.52
530
Chapter 19
◆
Ratio, Proportion, and Variation
7. If y varies inversely as the square root of x, fill in the missing values.
x
3567 5725
y 1136 1828
8. If y is inversely proportional to the cube root of x, by what factor will y change when x is
tripled?
9. If y is inversely proportional to the square root of x, by what percentage will y change when
x is decreased by 50.0%?
10. Plot the curve y 5x2 for the domain 0 to 5.
11. Plot the function y 1x for the domain 0 to 10.
Boyle’s Law
The pascal (Pa) is the SI unit of
pressure. It equals 1 newton per
square metre.
Boyle’s law states that for a confined gas at a constant temperature, the product of the pressure
and the volume is a constant. Another way of stating this law is that the pressure is inversely
proportional to the volume or that the volume is inversely proportional to the pressure. Assume
a constant temperature in the following problems.
12. A certain quantity of gas, when compressed to a volume of 2.50 m3, has a pressure of 184
Pa. Find the pressure resulting when that gas is further compressed to 1.60 m3.
13. The air in the cylinder in Fig. 19–4 is at a pressure of 101 kPa and occupies a volume of
2870 cm3. Find the pressure when it is compressed to 410 cm3.
14. A balloon contains 320 m3 of gas at a pressure of 140 000 Pa. What would the volume be
if the same quantity of gas were at a pressure of 250 000 Pa?
Gravitational Attraction
Newton’s law of gravitation states
that any two bodies attract each
other with a force that is inversely
proportional to the square of the
distance between them.
15. The force of attraction between two certain steel spheres is 375 pN (1 piconewton = 1012
N) when the spheres are placed 18.0 cm apart. Find the force of attraction when they are
52.0 cm apart.
16. How far apart must the spheres in problem 15 be placed to cause the force of attraction
between them to be 3.00 pN?
17. The force of attraction between the earth and some object is called the weight of that
object. The law of gravitation states, then, that the weight of an object is inversely proportional to the square of its distance from the centre of the earth. If a person weighs 150 lb.
on the surface of the earth (assume this to be 3960 mi. from the centre), how much will he
weigh 1500 mi. above the surface of the earth?
_
18. How much will a satellite, whose weight on earth is 675 N, weigh at an altitude of 250 km
above the surface of the earth?
Illumination
The inverse square law states
that for a surface illuminated
by a light source, the intensity
of illumination on the surface
is inversely proportional to the
square of the distance between
the source and the surface.
19. A certain light source produces an illumination of 800 lx (a lux [lx] is 1 lumen per square
metre) on a surface. Find the illumination on that surface if the distance to the light source
is doubled.
20. A light source located 2.75 m from a surface produces an illumination of 528 lx on that
surface. Find the illumination if the distance is changed to 1.55 m.
21. A light source located 7.50 m from a surface produces an illumination of 426 lx on_that
surface. At what distance must that light source be placed to give an illumination of 850 lx?
1
22. When a document is photographed on a certain copy stand, an exposure time of s is
25
needed, with the light source 0.750 m from the document. At what distance must the light
1
be located to reduce the exposure time to s?
100
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.