1. 3.7 Optimization Problems
Buffalo Bill’s Ranch, North Platte, Nebraska
Greg Kelly, Hanford High School, Richland, Washington
Photo 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 in2. 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 in2. 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. Maximum volume occurs with dimensions 6"×6"×3“.
A manufacturer wants to design an open box having a square base and a surface area of 108 in2. 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 1st or 2nd Derivative Test.
Rel max
Maximum volume occurs with dimensions 6"×6"×3“.

7. Which points on the graph of y = 4 – x2 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 – x2 are closest to the point (0, 2)?
3. Find critical numbers.
To minimize d, you can just minimize d2.
4. Use 1st or 2nd Derivative Test.
Closest points:
Rel max
Rel min
Rel min

9. A rectangular page is to contain 24 in2 of print
A rectangular page is to contain 24 in2 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. 223)
#1e
5 – 17 odd,
21, 23c