What dimensions will result in a box with the largest possible volume if an open rectangular box with square base is to be made from 48 ft^2 of material?

1 Answer
Feb 22, 2015

The base will be 4 xx 4 and the height will be 2 (all numbers in feet)

Let b be the length and the width of the base (length and width are the same since the base is square).

Let h be the height of the box.

The surface area of the box is
base + height xx perimeter
= b^2 + 4bh = 48
From which we can determine:
h = (48 - b^2)/(4b)

The Volume of the box:
V(b) = b^2h = b^2 * ((48 - b^2)/(4b))
= 12b - (3b^3)/4

The Volume is a maximum when ( d V(b))/ (db) = 0

(d V(b))/(db) = 12 - (3 b^2)/4 = 0

b = +-4 (only +4 is not extraneous)

Plugging this back into the formula
h = (48 - b^2)/(4b)
we get
h = 2