Question

A rectangular box with no top is to be built from 1728 square meters of material. Find the dimensions of such a box that will enclose the maximum volume.

Answer 1

given surface area with no top as 1728

We say

V= X x Y x Z

So

yz = a(y+2z)

xz = a(x+2z)

Then using partial derivatives

xy = a(2x+2y)

xy+2xz+2yz = 1728

So

xy+2yz = xy+2xz

x=y

So

4xz = x²+ 2xz

x = 2z

4z²+4z²+4z²= 1728

z² = 432

z = 20.8m

x = y = 41.6m