College Algebra 9th txtbk

SECTION 4–5 Variation and Modeling 319
EXAMPLE 4
Combined Variation
The frequency f of a vibrating guitar string is directly proportional to the square root of the
tension T and inversely proportional to the length L. What is the effect on the frequency if
the length is doubled and the tension is quadrupled?
SOLUTION
The equation of variation has the form
1T
f k
L
Let f 1 , T 1 , and L 1 denote the initial frequency, tension, and length, respectively. Then
L 2 2L 1 and T 2 4T 1 . Therefore,
f 2 k
1T 2
L 2
14T 1
k
2L 1
21T 1
k
2L 1
Let L 2 2L 1 , and T 2 4T 1 .
Simplify the radical.
Cancel and use the equation of variation.
f 1
We conclude that there is no effect on the frequency—the pitch remains the same.
MATCHED PROBLEM 4
Refer to Example 4. What is the effect on the frequency if the tension is increased by a
factor of 4 and the length is cut in half ?
ZZZ EXPLORE-DISCUSS 1
Refer to the equation of variation in Example 4. Explain why the frequency f, for fixed
T, is a rational function of L, but f is not, for fixed L, a rational function of T.
ANSWERS TO MATCHED PROBLEMS
1. k 25; p 25v 2. k 196; P 196 3. k 1; V x 2 h
V
4. The frequency is increased by a factor of 4.
4-5 Exercises
1. Suppose that y is directly proportional to x and that the constant
of proportionality is positive. If x increases, what happens to y?
Explain.
2. Suppose that y is directly proportional to x and that the constant
of proportionality is negative. If x increases, what happens to y?
Explain.
3. Suppose that y is inversely proportional to x and that the constant
of proportionality is positive. If x increases, what happens
to y? Explain.
4. Explain what it means for w to be jointly proportional to x and y.
5. Suppose that y varies directly with x. What is the value of y
when x 0? Explain.