1. 4.7 Optimization Problems

2. A Classic Problem
You have 40 feet of fence to enclose a rectangular garden along the side of a barn. What is the maximum area that you can enclose?

3. Optimization Problem
1. Assign variables and draw a figure/diagram, if possible.
2. Write the condition/constrain for the situation
3. Write an equation that relates the quantity to be optimized to other variables.
4. Use the conditions of the problem to write the quantity to be optimized as a function of ONE variable.
5. Find the feasible domain of that function
6. Find the absolute extrema of the function on the specified domain.

4. Examples
1. A television manufacturing firm needs to design an open-topped box with a square base. The box must hold 32 cu. in. Find the dimensions of a box that can be built with the minimum amount of materials.
A fence must be build to enclose a rectangular
area of 20,000 sq. ft. Fencing material costs $2.50
per foot for the two sides facing north and south,
and $3.20 per foot for the other two sides. Find the
cost of the least expensive fence.

5. Example
Two posts, one 12 feet high and the other 28 feet high, stand 30 feet apart. They are to be stayed by two wires, attached to a single stake, running from ground level to the top of each post. Where should the stake be placed to use the least amount of wire?
Note: This problem involves a lot of algebra. You can use technology to help solve the problem.

6. Group work
Find the dimensions of the rectangular field of maximum area that can be made from 300 m of fencing material.
25 m
50 m
125 m
100 m
10 m
75 m
140 m
75 m