7. Applied Maximum and Minimum Problems

by M. Bourne

The process of finding maximum or minimum values is called optimisation. We are trying to do things like maximise the profit in a company, or minimise the costs, or find the least amount of material to make a particular object.

These are very important in the world of industry.

Example 1

The daily profit, P, of an oil refinery is given by

P = 8x − 0.02x2,

where x is the number of barrels of oil refined. How many barrels will give maximum profit and what is the maximum profit?

[Go here to see another way to find the maximum or minimum value of a parabola.]

Continues below

Example 2

A rectangular storage area is to be constructed along the side of a tall building. A security fence is required along the remaining 3 sides of the area. What is the maximum area that can be enclosed with `800\ "m"` of fencing?

Example 3

[This problem was presented for discussion earlier in the Differentiation introduction.]

A box with a square base has no top. If 64 cm2 of material is used, what is the maximum possible volume for the box?