Answer to Question #167204 in Calculus for John

Question

A private shipping company will accept a box for domestic shipment only if the sum of its length and girth (distance around) does not exceed 108 in. Suppose you want to mail a box with square sides so that its dimensions are   and it's girth is   What dimensions will give the box its largest volume?

Answer 1

First, we will need to complete the question.

Let us assume that its dimensions are h by h by w and its girth is 2h + 2w. 

$Volume=h^2w$

Length + girth =108

$w+(2h+2w)=108$

$2h+3w=108 \implies w = \frac{108-2h}{3} \implies w = 36-\frac{2h}{3}$

$V=h^2w=h^2(36-\frac{2h}{3})=36h^2-\frac{2h^3}{3}$

Maximum volume $\frac{dV}{dh}=0 \implies\frac{dV}{dh}=0=72h-2h^2 \implies h=36$

$w=36−\frac{2}{3}*36=12$

The dimension that gives the box its largest volume is $36\times36\times12$