1. Mathematical Modeling & Variation MATH 109 - Precalculus S. Rook

2. Overview Section 1.10 in the textbook: – Direct variation – Direct variation as an n th power – Inverse variation – Combined variation – Joint variation 2

3. Direct Variation

4. Variation in General Given two quantities that are related, variation refers to how either increasing or decreasing the first quantity affects the second quantity Because the two quantities are related, they differ by only a constant value – This constant is called the constant of proportionality and is often denoted by k We can model variation by equations Variation can occur with more than two quantities – Will see this later 4

5. Direct Variation Direct variation: situations that can be modeled with the formula y = kx where x and y represent the two quantities k is the constant of proportionality The following statements are all equivalent and indicative of direct variation (also in the book): y varies directly as x y is directly proportional to x y = kx for some nonzero constant k 5

6. Direct Variation (Example) Ex 1: Solve by setting up a mathematical model: a)It is observed that 33 centimeters is approximately equal to 13 inches. Given that the number of centimeters vary directly with the number of inches, find how many centimeters are in 20 inches b)Hooke’s law states that the distance a spring is stretched or compressed from its starting point is directly proportional to the force exerted on the spring. If 100 Newtons stretches a spring 0.25 meters, how far will a force of 80 Newtons stretch the spring? 6

7. Direct Variation as an n th Power Same as direct variation except x is raised to a power y = kx n x and y represent the two quantities k is the constant of proportionality The following statements are all equivalent and indicative of direct variation as an n th power (also in the book): y varies directly as the n th power of x y is directly proportional to the n th power of x y = kx n for some nonzero constant k 7

8. Direct Variation as an n th Power (Example) Ex 2: Solve by setting up a mathematical model: Suppose that the surface area of a box varies directly as the square of the measure of its sides. If a box with sides measuring 4 inches has a surface area of 96 square inches, find the measure of the sides of a box with a surface area 864 square inches 8

9. Inverse Variation

10. Inverse variation: situations that can be modeled with the formula y = k ⁄ x where x and y represent the two quantities k is the constant of proportionality The following statements are all equivalent and indicative of inverse variation (also in the book): y varies inversely as x y is inversely proportional to x y = k ⁄ x for some nonzero constant k 10

11. Inverse Variation (Example) Ex 3: Solve by setting up a mathematical model: Boyle’s law states that for a constant temperature, the pressure of a gas is inversely proportional to the volume of the gas. Suppose a gas occupying 4 liters exerts a pressure of 6 atmospheres. How much pressure would be exerted from a gas occupying 10 liters? 11

12. Combined Variation

13. Combined variation: situations that can be modeled with the formula z = kx ⁄ y where x and y, and z represent the quantities k is the constant of proportionality The following statements are all equivalent and indicative of combined variation: z varies directly as x and inversely as y z is directly proportional to x and inversely proportional to y z = kx ⁄ y for some nonzero constant k 13

14. Combined Variation (Example) Ex 4: Solve by setting up a mathematical model: z varies directly as the square of x and inversely as y. If z = 6 when x = 6 and y = 4, find z when x = 2 and y = 8 14

15. Joint Variation

16. Joint variation: situations that can be modeled with the formula z = kxy where x and y, and z represent the quantities k is the constant of proportionality The following statements are all equivalent and indicative of joint variation (also in the book): z varies jointly as x and y z is jointly proportional to x and y z = kxy for some nonzero constant k 16

17. Joint Variation (Example) Ex 5: Solve by setting up a mathematical model: The work done when lifting an object varies jointly with the mass of the object and the height that the object is lifted. The work done when a 120-kilogram object is lifted 1.8 meters is 2116.8 joules. How much work is done when lifting a 100-kilogram object 1.5 meters? 17

18. Summary After studying these slides, you should be able to: – Solve problems modeling direct variation, direct variation to the n th power, inverse variation, combined variation, and joint variation Additional Practice – See the list of suggested problems for 1.10 Next lesson – Quadratic Functions and Models (Section 2.1) 18