1. Section 3.5 Mathematical Modeling Objective(s): To learn direct, inverse and joint variations. To learn how to apply the variation equations to find the constant of variation. To learn how to find the mathematical answer after inputting the given constraints and substituting the found constant of variation.

2. HW 31-36: Write a mathematical model for each of the following: A) y varies directly as the cube of x B) h varies inversely as the square root of s Example 1:

3. HW 31-36 Cont: Write a mathematical model for each of the following: D) Suppose x varies directly as y and inversely as z. C) c is jointly proportional to the square of x and Example 2:

4. HW 47-54: Write a mathematical model for each of the following. In each case, determine the constant of proportionality. A) y varies directly as the cube of x. (y = 81 when x = 3) B) h varies inversely as the square root of s. (h = 2 when s = 4) C) c is jointly proportional to the square of x and (c = 144 when x = 3 and y = 2) Example 3:

5. The stopping distance d of an automobile is directly proportional to the square of its speed s. A car required 75 feet to stop when its speed was 30 mph. Estimate the stopping distance if the brakes are applied when the car is traveling at 50 mph. Example 4:

6. Direct Variation - Inverse Variation - x 5 10 15 20 y 24 12 8 6 If direct, then y = kx 24 = k(5) If inverse, then k = 120 INVERSE Example 5:

7. Homework: Pg. 309(13-15) all, (31-36) all, (47-54) all

8. The distance a spring is stretched (or compressed) varies directly as the force on the spring. A force of 220 newtons stretches a spring 0.12 meters. What force is required to stretch the spring 0.16 meters? Example 6: (Application problem)