A company wants to design a box with four rectangular sides and a square top and bottom.

a) Letting denote the length of one of the sides of the bottom of the box, and letting denote the height of the box, sketch a picture of the box, labeling the length, width, and height. The company wants to use exactly 121 cm of material for the box. That is, the box's surface area will be equal to 121 cm. Write an equation in terms of and representing this constraint, putting the variables on the left side.

Solve for to find as a function of .

b) Using your work from part (a), write a formula for the volume of the box, , as a function of . Simplify as much as possible.

c) Use an online source to approximate the maximum value of on the interval . What value corresponds to this maximum value?

What does this mean in the context of the problem?