Chapter 2.1, Problem 27E

A Precocious Child and Her Blocks A child has 64 blocks that are 1-inch cubes. She wants to arrange the blocks into a solid rectangle h blocks long and w blocks wide. There is a relationship between h and w that is determined by the restriction that all 64 blocks must go into the rectangle. A rectangle h blocks long and w blocks wide uses a total of h X w blocks. Thus, hw = 64. Applying some elementary algebra, we get the relationship that we need:

$w=\frac{64}{h}\cdots \left(2.3\right)$

1. Use a formula to express the perimeter P in terms of h and w.
2. Using Equation (2.3), find a formula that expresses the perimeter P in terms of the height only.
3. How should the child arrange the blocks if she wants the perimeter to be the smallest possible?
4. Do parts b and c again, this time assuming that the child has 60 blocks rather than 64 blocks. In this situation, the relationship between h and w is w = 60/h. {Note: Be careful when you do part c. The child will not cut the blocks into pieces!)