This document discusses different types of variation used in modeling: 1) Direct variation where one quantity varies directly as another quantity. An example given is volume of a sphere varying directly as the cube of the radius. 2) Inverse variation where one quantity varies inversely as another. An example is pressure of a gas varying inversely as the volume of its container. 3) Combined variation where quantities can vary directly and inversely with other quantities. An example problem is given to write an equation for this type of variation. 4) Joint variation where a quantity varies jointly as two other quantities. An example problem is given to find the volume of a pyramid with given height and base dimensions.

1. Section 3.7 Modeling Using Variation

2. Direct Variation

5. Example The volume of a sphere varies directly as the cube of the radius. If the volume of a sphere is 523.6 cubic inches when the radius is 5 inches, what is the radius when the volume is 33.5 cubic inches. r

6. Inverse Variation

8. Example The pressure, P, of a gas in a spray container varies inversely as the volume, V, of the container. If the pressure is 6 pounds per square inch when the volume is 4 cubic inches, what is the volume when the pressure is down to 3 pounds per square inch?

9. Combined Variation

11. Example The TIXY calculator leasing company has determined that leases L, vary directly as its advertising budget and inversely as the price/month of the calculator rentals. When the TIXY company spent $500 on advertising on the internet and charge $30/month for the rentals, their monthly rental income was $4000. Write an equation of variation that describes this situation. Determine the monthly leases if the amount of advertising is increased to $2000.

12. Joint Variation

13. d m 1 m 2

14. Example The volume of a model square based pyramid, V, various jointly as its height, h, and the square of its side, s ,of the square base. A model pyramid that has a side of the square base that is 6 inches, and the height is 10 inches, has a volume of 120 cubic inches. Find the volume of a pyramid with a height of 9 inches and a square base of 5 inches.

15. (a) (b) (c) (d) r

16. (a) (b) (c) (d)