Suppose you have 100 inches of wire to form the skeleton of a rectangular box with a square base. What dimensions of the box will maximize the volume?
The formula for volume for the box is area of the base times the height. Side the base is square, the area of the base is x^2, with x being the length of a side of the square. Let's make the height "y", therefore the volume is (x^2)y.
The distance around the base is 4x since each side is of length "x", therefore it is 8x around the top and bottom of the box. The sides are each length "y", so the four sides have length 4y.
So we know that 8x + 4y = 100.
Now solve the equation for either x or y. Solving for y, we get y = (100 - 8x)/4 = 25 - 2x.
Substitute that in for the volume equation to get volume = x^2(25 - 2x) = 25x^2 - 2x^3. To maximize the volume, take the derivative of the volume equation, which equals 50x - 6x^2.
Set this equal to zero and solve for x to get, 2x(25 - 3x) = 0. Therefore x = 0 or 25/3.
The dimensions of the box are 25/3 by 25/3 by 25/3, so the box is a cube.