• 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 wh2. 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!