Question

According to postal regulations, a carton is classified as "oversized" if the sum of its height and girth (the perimeter of its base) exceeds 108 in. Find the dimensions of a carton with square base that is not oversized and has maximum volume.

Answer 1

Answer:

18" X 18" X 36"

Step-by-step explanation:

Given a square base container of height h, let a side of the base =s

The volume of the container, $V= s^2h$

If the sum of its height and girth (the perimeter of its base) equals 108 in

$4s+h = 108\\h = 108-4s$

Substituting h=108-4s into V

$V= s^2(108-4s)$

We are required to determine the maximum volume of such container, first we take the derivative:

$V'(s)= s(216-12s)$

Optimizing:

$s(216-12s)=0\\216-12s=0\\12s=216\\s=216 \div 12\\s=18$

Recall that: h = 108-4s

$h = 108-4(18)=108-72=36inch$

The dimensions of the carton are 18" X 18" X 36".