What is the maximum possible volume, in cubic inches, of the box?

Question

An open-top box is to be made from a 30-inch by 42-inch piece of plastic by removing a square from each corner of the plastic and folding up the flaps on each side. What is the maximum possible volume, in cubic inches, of the box? You may enter an exact answer or round to the nearest hundredth.

Answer 1

Let x be the length of a side of the square. Now, when you form a box, the sides of the base will be (30-2x) and (42-2x) and the height will be x.

You can now determine the volume in terms of x (length x width x height)

You can find dV/dx and set it to 0 to find critical points.

Plug in solution to V expression to find max V. Look for sign change from + to - in dV/dx for max.