- Problem 5.5 #5:
Suppose a closed box has a fixed area (including the four sides, the top
and the bottom), A. Say the bottom is a square, x feet long on a side.
What's the maximum volume such a box could have?
- Problem 5.5 #8:
Start with a log of radius 30 cm. We want to mill it
so that it has a rectangular cross section, h cm high and w cm wide.
The strength of such a beam is proportional to wh
^{2}.
Find the values of w and h that maximize the strength of the beam.

General Procedure:
- [1] Read the problem (draw a picture, perhaps, to help understand the problem).
- [2] Determine what quantity Q is to be optimized.
- [3] Find a formula for Q in terms of variables used in the problem.
- [4] Pick one variable (some choices may be better than others!) and
express the other variables in terms of the one you picked.
- [5] Substitute into your formula for Q so that Q becomes a function
of only one variable (the one you picked).
- [6] Determine what range of values of your variable is appropriate
for the problem.
- [7] Optimize Q over this range of values, using the methods of section 5.3.
- [8] Be sure to answer the question that was asked!