Find volume?

Question

An open box is to be made from a twenty-four-inch by twenty-four-inch square piece of material by cutting equal squares from the corners and turning up the sides (see figure below, where y = 24 − 2x). Find the volume of the largest box that can be made.

Answer 1

From the given data, each cut - out measures x inches by x inches, the dimensions of the box are x by (24-2x) by (24-2x).

Means that,

Length of the box (l) = x,

Width of the box (w) = 24 - 2x, and

Height of the box (h) = 24 - 2x.
Volume of the box (V) = l * w * h

= x * (24 - 2x) * (24 - 2x)

= x(24-2x)²

= (4x³ - 96x² + 576x) in³.

To find the maximum volume, find the derivative of 4x³ - 96x² + 576x and then, equating it to zero.

12x² - 192x + 576 = 0
x² - 16x + 48 = 0
x = [16 ± √(16² - 4*1*48)]/(2*1)
= [16 ± √64]/2
= [16 ± 8]/2
= 4, 12.

when, x = 12, w = 24 - 2x = 24 - 24 = 0.

when, x = 12, h = 24 - 2x = 24 - 24 = 0.
So, consider x = 4.

Length of the box (l) = x = 4.

Width of the box (w) = 24 - 2x = 24 - 8 = 16.

Height of the box (h) = 24 - 2x = 24 - 8 = 16.
Volume of the box (V) = l * w * h

= 4 * 16 * 16

= 1024 in³ .

Volume of the largest box is 1024 in³ .