A box of rectangular base and an open top has a surface area 600 cm2. If the height of the box is equal to its width, find the dimensions that give a maximum volume.

Answer 1

With an open top, the surface area is lw + 2lh + 2wh (only 1 lw because of the open top).

l = w, so surface area is lw + 2lw + 2w² = 3lw + 2w²

maximize lwh = lw² subject to 3lw + 2w² = 600

There are many ways to handle this. One is to first solve for l in terms of w. l = (600 - 2w²)/3w = 200/w - 2/3 w

Then, lw² = (200/w - 2/3 w)w² = 200w - 2/3w³

We maximize by finding where the derivative is 0.

200 - 2 w² = 0

w = 10

l = 200/w - 2/3 w = 200/10 - 2/3 * 10 = 20 - 6 2/3 = 13 1/3 or 40/3

lwh = 40/3 * 10 * 10 = 4000/3 = 1333 1/3 cm³

If you want to solve this using Excel, you can use the solver and maximize lw² subject to 3lw + 2w² = 600