Hawkes Learning Systems: College Algebra

1. Hawkes Learning Systems:College Algebra Section 4.3b: Direct and Inverse Variation

2. Objective • Variation problems.

3. Variation Problems Direct Variation We say that y varies directly as the n ͭ ͪ power of x(or that y is directly proportional to the n ͭ ͪ power of x) if there is a non-zero constant k (called the constant of proportionality) such that

4. Variation Problems Inverse Variation We say that y varies inversely as the n ͭ ͪ power of x(or that y is inversely proportional to the n ͭ ͪ power of x) if there is a non-zero constant k such that

5. Example: Variation Problems For the following phrases, write the general formula that applies. • “y varies inversely as the n ͭ ͪpower of x” • “y is directly proportional to the n ͭ ͪpower of x” • “y is inversely proportional to the n ͭ ͪpower of x” d. “y varies directly as the n ͭ ͪpower of x”

6. Example: Variation Problems The weight of a person, relative to the Earth, is inversely proportional to the square of the person’s distance from the center of the Earth. Using a radius for the Earth of 6370 kilometers, how much does a 240 lb man weigh when flying in a jet 11 kilometers above the Earth’s surface? Continued on the next slide…

7. Example: Variation Problems (cont.) If we let W stand for the weight of the person and let d stand for the distance between the person and the Earth’s center, we know that . We also know that the W = 240 lbswhen d = 6370 km. Continued on the next slide…

8. Example: Variation Problems (cont.) Solving the equation for k and substituting in the values that we know, we obtain When the man is 11 km above the Earth’s surface, d=6381, so the man’s weight while flying is:

9. Example: Variation Problems Hooke’s law says that the force exerted by the spring in a spring scale varies directly with the distance that the spring is stretched. If an 8 pound mass suspended on a spring scale stretches the spring 3 inches, how far will a 12 pound mass stretch it? Continued on the next slide…

10. Example: Variation Problems (cont.) We know that F =kxwhereF represents the force exerted by the spring andxrepresents the distance that the spring is stretched. The spring exerts a force of 8 pounds because the 8 pound weight is suspended on it. Continued on the next slide…

11. Example: Variation Problems (cont.) So, or Then, Thus, the spring stretches 4.5 inches when a 12 pound mass is suspended from it.