Find the maximum volume of a rectagular box with square ends that satisfies the delivery company's requirement?

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 112 inches. Find the maximum volume of a rectangular box with square ends that satisfies the delivery company's requirements.Find the maximum volume of a rectagular box with square ends that satisfies the delivery company's requirement?
Equations involved:

4s + L = 112.......where s is length of one side of square end, L = length of box

L = 112 - 4s........Now we go to volume

V = (s^2)(L).......replacingg L with relationship above

V = s^2(112 - 4s).........expanding

V = 112s^2 - 4s^3........now the derivative will show the change in volume relative to change in s, and the maximum will be at 0 (top of this curve)...so take the derivative and set it equal to zero

0 = 224s - 12s^2

0 = s(224 - 12s)......so s = zero (throw that out) and

0 = 224 - 12s

12s = 224

s = 224/12

s = 18.7

L = 112 - 4s

L = 37.2...check 37.2 + 4(18.7) = 112

V = 37.2(18.7)^2

V = 13,008.5............answer



Hope that makes sense