A closed rectangular box

Question

A (closed)rectangular box with a square base has a surface area of 150 square inches. Find the dimensions that give the maximum volume. What is the maximum volume?

(Hint: The surface area of a closed rectangular box with a square base of length x and width x and height h is given by the formula A = 2x2 + 4xh and the volume is given by the formula V = x2h.)

Answer 1

Typically this type of problem (constrained maximization) is solved using Lagrange Multipliers. I am guessing, however, that that isn't how you're expected to do it. If I'm wrong, let me know. To do it without Lagrange Multipliers, let's look at the constraint (fixed surface area):

150 = 2x^2 + 4xh

Let's solve this for h: h = (150 - 2x^2)/4x

and plug this into the equation for volume:

V = x^2h = x(150 - 2x^2)/4

Taking the derivative with respect to x and setting it equal to zero:

(150 - 2x^2)/4 - x^2 = 0

Which equates to x = 5.

Plug this into h = (150 - 2x^2)/4x and h = 5 too.

So all dimensions are 5 and the rectangular box is actually cubic.