Chapter 6.2, Problem 29E

OTHER APPLICATIONS

Container Design Consider the problem of cutting corners out of a rectangle and folding up the sides to make a box. Specific examples of this problem are discussed in Example 3 and Exercise $28$.

a. In the solution to Example 3, compare the area of the base of the box with the area of the walls.

b. Repeat part a for the solution to Exercise $28$.

c. Make a conjecture about the area of the base compared with the area of the walls for the box with the maximum volume.

EXAMPLE 3 Maximizing Volume

An open box is to be made by cutting a square from each corner of a $12-\text{in}\text{.}$ by $12-\text{in}\text{.}$ piece of metal and then folding up the sides. What square size should be cut from each corner to produce a box of maximum volume?

28. Container Design An open box will be made by cutting a square from each corner a $3-\text{ft}$ by $8-\text{ft}$ piece of cardboard and then folding up the sides. What size square should be cut from each corner in order to produce a box of maximum volume?