Section 19–2
◆
515
Direct Variation
28. A line, as shown in Fig. 19–2, is subdivided into two segments a and b such that the ratio
of the smaller segment a to the larger segment b equals the ratio of the larger b to the whole
(a b). The ratio ab is called the golden ratio or golden section. Set up a proportion,
based on the above definition, and compute the numerical value of this ratio.
a
FIGURE 19–2
b
A line subdivided by the golden ratio.
19–2 Direct Variation
Constant of Proportionality
If two variables are related by an equation of the following form:
Direct
Variation
y kx
or
Glance back at Sec. 5–3 where
we plotted the equation (Eq.
289) of straight line
yx
y mx b
60
where k is a constant, we say that y varies directly as x, or that y is directly proportional to x.
The constant k is called the constant of proportionality. As shown above, direct variation may
also be written using the special symbol ⴤ:
yx
It is read “y varies directly as x” or “y is directly proportional to x.”
Solving Variation Problems
Variation problems can be solved with or without evaluating the constant of proportionality. We
first show a solution in which the constant is found by substituting the given values into Eq. 60.
◆◆◆
Example 8: If y is directly proportional to x, and y is 27 when x is 3, find y when x is 6.
Solution: Since y varies directly as x, we use Eq. 60.
y kx
To find the constant of proportionality, we substitute the given values for x and y, 3 and 27.
27 k(3)
So k 9. Our equation is then
y 9x
When x 6,
y 9(6) 54
◆◆◆
We now show how to solve such a problem without finding the constant of proportionality.
◆◆◆
Example 9: Solve Example 8 without finding the constant of proportionality.
Solution: When quantities vary directly with each other, we can set up a proportion and solve
it. Let us represent the initial values of x and y by x1 and y1, and the second set of values by x2
and y2. Substituting each set of values into Eq. 60 gives us
y1 kx1
We see that Eq. 60 is the
equation of a straight line with
a slope of k and a y intercept
of zero.
516
Chapter 19
◆
Ratio, Proportion, and Variation
and
The same proportion can also be
written in the form y1/x1 y2/x2.
Two other forms are also possible:
x1
y1
x2
y2
and
y2 kx2
We divide the second equation by the first, and k cancels.
y2 x2
y1 x1
The proportion says, The new y is to the old y as the new x is to the old x. We now substitute the
old x and y (3 and 27), as well as the new x (6).
x1 x1
y2 y2
y2
y1
6
3
Solving yields
6
y2 27 p 3
◆◆◆
q
54 as before
◆◆◆
Example 10: If y varies directly as x, fill in the missing numbers in the table of values.
x
1
2
y
5
16
20
28
Solution: We find the constant of proportionality from the given pair of values (5, 20).
Starting with Eq. 60, we have
y kx
and substituting gives
20 k(5)
k4
So
y 4x
With this equation we find the missing values.
When x 1:
y4
When x 2:
y8
16
x 4
4
28
x 7
4
When y 16:
When y 28:
So the completed table is
Unstretched
length
F
x
FIGURE 19–3
x
1
2
4
5
7
y
4
8
16
20
28
◆◆◆
Applications
Many practical problems can be solved using the idea of direct variation. Once you know
that two quantities are directly proportional, you may assume an equation of the form of
Eq. 60. Substitute the two given values to obtain the constant of proportionality, which
you then put back into Eq. 60 to obtain the complete equation. From it you may find any
other corresponding values.
Alternatively, you may decide not to find k, but to form a proportion in which three
values will be known, enabling you to find the fourth.
Section 19–2
◆
517
Direct Variation
◆◆◆ Example 11: The force F needed to stretch a spring (Fig. 19–3) is directly proportional
to the distance x stretched. If it takes 15 N to stretch a certain spring 28 cm, how much force is
needed to stretch it 34 cm?
Estimate: We see that 15 N will stretch the spring 28 cm, or about 2 cm per newton. Thus a
stretch of 34 cm should take about 34 2, or 17 N.
Solution: Assuming an equation of the form
F kx
and substituting the first set of values, we have
15 k(28)
15
k 28
15
So the equation is F x. When x 34,
28
15
F p q 34 18 N
28
(rounded)
which is close to our estimated value of 17 N.
Exercise 2
◆
◆◆◆
Direct Variation
1. If y varies directly as x, and y is 56 when x is 21, find y when x is 74.
2. If w is directly proportional to z, and w has a value of 136 when z is 10.8, find w when z
is 37.3.
3. If p varies directly as q, and p is 846 when q is 135, find q when p is 448.
4. If y is directly proportional to x, and y has a value of 88.4 when x is 23.8:
(a) Find the constant of proportionality.
(b) Write the equation y f (x).
(c) Find y when x 68.3.
(d) Find x when y 164.
Assuming that y varies directly as x, fill in the missing values in each table of ordered pairs.
5.
6.
x
9
y
45
x
11
3.40 7.20
y
7.
x
75
12.3
50.4 68.6
115
y
125
154
167
187
8. Graph the linear function y 2x for values of x from 5 to 5.
Applications
9. The distance between two cities is 828 km, and they are 29.5 cm apart on a map. Find the
distance between two points 15.6 cm apart on the same map.
10. If the weight of 2500 steel balls is 3.65 kg, find the number of balls in 10.0 kg.
11. If 80 transformer laminations make a stack 1.75 cm thick, how many laminations are
contained in a stack 3.00 cm thick?
518
Chapter 19
Weight
Plunger
Compressed
gas
FIGURE 19–4
For problem 17, use Charles’ law:
The volume of a gas at constant
pressure is directly proportional
to its absolute temperature. K is
the abbreviation for kelvin, the SI
absolute temperature scale. Add
273.15 to Celsius temperatures
to obtain temperatures on the
kelvin scale.
◆
Ratio, Proportion, and Variation
12. If your car now gets 9.00 km/L of gasoline, and if you can go 400 km on a tank of
gasoline, how far could you drive with the same amount of gasoline in a car that gets
15.0 km/L?
13. A certain automobile engine delivers 53 kW and has a displacement (the total volume
swept out by the pistons) of 3.0 L. If the power is directly proportional to the displacement, what power would you expect from a similar engine that has a displacement of
3.8 L?
14. The resistance of a conductor is directly proportional to its length. If the resistance of
2.60 km of a certain transmission line is 155 , find the resistance of 75.0 km of that
line.
15. The resistance of a certain spool of wire is 1120 . A piece 10.0 m long is found to have a
resistance of 12.3 . Find the length of wire on the spool.
16. If a certain machine can make 1850 parts in 55 min, how many parts can it make in 7.5 h?
Work to the nearest part.
17. In Fig. 19–4, the constant force on the plunger keeps the pressure of the gas in the cylinder
constant. The piston rises when the gas is heated and falls when the gas is cooled. If the
volume of the gas is 1520 cm3 when the temperature is 302 K, find the volume when the
temperature is 358 K.
18. The power generated by a hydroelectric plant is directly proportional to the flow rate
through the turbines, and a flow rate of 5625 imperial gallons of water per minute produces 41.2 MW. How much power would you expect when a drought reduces the flow to
5000 gal./min?
19–3
The Power Function
Definition
In Sec. 19–2, we saw that we could represent the statement “y varies directly as x” by Eq. 60,
y kx
Similarly, if y varies directly as the square of x, we have
y kx2
or if y varies directly as the square root of x,
y k x kx1/2
These are all examples of the power function.
Power
Function
The constants can, of course,
be represented by any letter.
Appendix A shows a instead of k.
For exponents of 4 and 5, we have
the quartic and quintic functions,
respectively.
y kxn
196
The constants k and n can be any positive or negative number. This simple function gives us a
great variety of relations that are useful in technology and whose forms depend on the value of
the exponent, n. A few of these are shown in the following table:
When:
We Get the:
Whose Graph Is a:
n1
n2
Linear function
Quadratic function
y kx (direct variation)
y kx2
Straight line
Parabola
n3
Cubic function
y kx3
k
y (inverse variation)
x
Cubical parabola
n 1
Hyperbola