Applied Maximum and Minimum Problems

Optimization Problems

1. Draw an appropriate figure and label the quantities relevant to the problem.

2. Write a primary equation that relates the given and unknown quantities.

3. If necessary, reduce the primary equation to 1 variable (use a secondary equation if necessary).

4. Determine the desired max/min using the derivative(s).

5. Check solutions with possible values (domain).

Example: An open box is to be made from a 16” by 30” piece of cardboard by cutting out squares of equal size from the four corners and bending up the sides. What size should the squares be to obtain a box with the largest volume?

xx

16

30

Example: An offshore oil well located at a point W that is 5 km from the closest point A on a straight shoreline. Oil is to be piped from W to a shore point B that is 8 km from A by piping it on a straight line under water from W to some shore point P between A and B and then on to B via pipe along the shoreline. If the cost of laying pipe is $1 million under water and $½ million over land, where should the point P be located to minimize the coast of laying the pipe?

5 km

A P B

x 8 – x

8 km

W

abs max

Example: You have 200 feet of fencing to enclose two adjacent rectangular corrals. What dimensions should be used to maximize the area?

Example: Find the radius and height of the right circular cylinder of largest volume that can be inscribed in a right circular cone with radius 4 inches and height 10 inches.

r

h

10-h

abs max