1. 3.7 Optimization Problems Buffalo Bill’s Ranch, North Platte, Nebraska Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 1999

2. Objective Solve applied minimum and maximum problems.

3. When might someone be interested in finding minimum or maximum values? Minimizing cost Maximizing profit Optimizing package sizes Minimizing surface area (material) given a certain volume Maximizing volume given a certain material Maximizing pasture given a certain amount of fencing

4. A manufacturer wants to design an open box having a square base and a surface area of 108 in 2. What dimensions will produce a box with maximum volume? 1. Determine what you’re maximizing or minimizing. Find an equation. Maximize volume 2. Get an equation in terms of one independent variable.

5. A manufacturer wants to design an open box having a square base and a surface area of 108 in 2. What dimensions will produce a box with maximum volume? 3. Find critical numbers. Can’t be –6, so x=6. 4. Look at endpoints (if there are any).

6. A manufacturer wants to design an open box having a square base and a surface area of 108 in 2. What dimensions will produce a box with maximum volume? 5. If there are endpoints, plug critical numbers and endpoints into the original equation to determine min/max. max 6. If no endpoints use 1 st or 2 nd Deriviative Test. Rel max Maximum volume occurs with dimensions 6"×6"×3“.

7. Which points on the graph of y = 4 – x 2 are closest to the point (0, 2)? 1. Determine what you’re maximizing or minimizing. Find an equation. Minimize distance 2. Get an equation in terms of one independent variable.

8. Which points on the graph of y = 4 – x 2 are closest to the point (0, 2)? 3. Find critical numbers. To minimize d, you can just minimize d 2. 4. Use 1 st or 2 nd Derivative Test. Rel max Rel min Closest points:

9. A rectangular page is to contain 24 in 2 of print. The margins at the top and bottom of the page are to be 1.5 inches, and the margins on the left and right are to be 1 inch. What should the dimensions of the page be so that the least amount of paper is used?

10. Homework 3.7 (p. 216) #1-13 odd, 17, 19c, 20c