Optimization

The best choice

Cut equal squares from the corners of a 12 by 12 sheet and fold up the sides to make an open box. Small squares give a shallow box. Large squares give a narrow one. Which size gives the most volume?

Drag the slider to try different choices and watch the graph of the quantity you are optimizing. Hunt for the best value by hand first, then turn on the calculus answer and compare.

The method

1. Name the quantity

Decide what you are maximizing or minimizing, and write it as a formula. It usually has two or more variables at first.

2. Use the constraint

The fixed amount, such as the sheet size, the fence length, or the volume, lets you write one variable in terms of the other. Substitute to get a function of one variable.

3. Find the domain

Lengths must be positive. The domain often has endpoints that need checking too.

4. Differentiate and solve

Set the derivative to 0, find the critical numbers, and check which gives the best value, using the first derivative test or by comparing with the endpoints.

For the box, V(x)=x(12-2x)^2 on 0. Then V'(x)=(12-2x)(12-6x), which is 0 at x=2 and x=6. Only x=2 is inside the domain, and it gives V=128. Why does the derivative graph above cross 0 right where the volume peaks?

Common mistakes

Differentiating before using the constraint

A formula with two variables cannot be optimized with one derivative. Use the constraint first.

Reporting the critical number instead of the value

Read the question. "What is the largest volume" asks for V(2)=128, not x=2.

Ignoring the domain

A critical number outside the domain, such as x=6 for the box, is not a valid design.