The postal service has a limit of 108 inches on the combined

Solution

If the cross section is a square, then the dimensions around the box (\"girth\") can all be called x. Let\'s call the length of the box L. So we have 4x + L = 108, or L = 108-4x. The volume of the rectangular box is V = x*x*L = x²(108-4x) = -4x³ + 108x². I don\'t know if you\'re in calculus, but if so, find the maximum of this function using the derivative. If you\'re not in calculus, you can find the maximum of the function using a graphing calculator. Calculus method: V \'(x) = -12x² + 216x 0 = -12x² + 216x 0 = -12x (x - 18) x = 0 or x = 18. Clearly x=0 makes no sense in the context of the problem, leaving x=18 as the possible maximum. Since f\'(1) >0 and f \'(20) <0, the function is indeed increasing and then decreasing on either side of x=18, so there is a maximum volume when x=18. The dimensions of the box are therefore 18 x 18 x 36 inches.