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 120 in. What dimensions will give a box with a square end the largest possible​ volume?

Answer 1

20 in, 40 in

Explanation:

Let L is the length and W is the width.

Total distance around, P = 2 L + 2 W = 120

L + W = 60 ..... (1)

Volume of box, V = W² x L

V = W² x (60 - W) from equation (1)

V = 60 W² - W³

dV/dW = 120 W - 3W²

It is equal to zero for maxima and minima

120 W - 3 W² = 0

W = 40 in

L = 60 - 40 = 20 in

Thus, the dimensions of the box is L = 20 in, W = 40 in