A parcel delivery service will deliver a package only if the length plus the girth (distance around, taken perpendicular to the length) does not exceed 100 inches. Find the maximum volume of a rectangular box with square ends that satisfies the delivery company's requirements.

Answer 1

let the square have side s
let the length be h

Let us assume that we use the maximum possible dimensions. That is,

4s + h = 100
v = hs^2 = (100-4s)s^2 = 100s^2 - 4s^3
dv/ds = 200s - 12s^2 = 4s(50-3s)

dv/ds = 0 when s = 50/3, h=100/3

v(50/3) = 250000/27 = 9259.259