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.

Answer 1

4 s + L = 112 ... L = 112 - 4 s

v = s^2 * L = s^2 (112 - 4 s) = - 4 s^3 + 112 s

find the local max v

Answer 2

Let the width and height be x inches each, (you said the ends are square)
let the length be y inches
your condition: y + 4x < 112
y < 112 - x

V = x^2 y = x^2(112-x) = 112x^2 - x^3
dV/dx = 224x - 3x^2 = 0 for a max
3x^2 = 224x
x = 224/3 , then y < 112/3

sub into V = x^2 y to get the maximum volume

Answer 3

a similar question...
https://www.jiskha.com/questions/621171/A-parcel-delivery-service-will-deliver-a-package-only-if-the-length-plus-the-girth