Optimization Problem...?

Question

I am confused as to how to go about this problem. We are studying optimization in calculus. Thank you for any help!

If 1200 cm^2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.

Answer 1

volume of the box =a^3
area of a cube=6a^2
area of the box to be made with open top=6a^2-a^2=5a^2

Answer 2

You should always start a problem like this by defining appropriate variables.

Let b denote the length (in cm) of the square base of the box.
Let h denote the height (in cm) of the box.

The surface area consists of the base, of area b², and the four vertical sides, each of area bh. So the total area A is

A = b² + 4bh

This is to equal 1200, so we have

b² + 4bh = 1200

The volume V of a rectangular box is length × width × height; so here

V = b²h

Solve the condition b² + 4bh = 1200 for h, substitute that for h in the expression for volume, and proceed as usual (solve dV/db=0 for b)