Question

A private shipping company will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 114 inches. what dimensions will give a box with a square end the largest possible volume?

Answer 1

Let 4x = the girth (each side = x) Let y = the length 
4x + y = 114 maximum y = 114 - 4x 
volume = x^2 * y v = x^2 * (114 - 4x) v = 114x^2 - 4x^3 . . . . . . v is maximum (or minimum) when it's derivative = 0 v ' = 228x - 12x^2 
228x - 12x^2 = 0 12x * (19 - x) = 0 
x = 0 <=== this is obviously a minimum volume or x = 19 inches <=== for maximum volume 
4x + y = 1144*19 + y = 114y = 38 <=== 
The largest volume is 19 x 19 x 38