What is the volume of the largest box that can be made from a square piece of cardboard with side lengths of 24 inches by cutting equal squares from each corner and turning up the sides?

1024 in³

Step-by-step explanation:

Volume = (24 - 2x) (24 - 2x) (x)

= 576x - 96x² + 4x³

dV/dx = 576 - 192x + 12x² = 0

x² - 16x + 48 = 0

x² - 12x - 4x + 48 = 0

x (x - 12) - 4 (x - 12) = 0

(x - 12) (x - 4) = 0

x = 4, x = 12

x can not be 12,

Because 24 - 2 (12) = 0

So x = 4

Volume = (24 - 8) (24 - 8) (4)

= 1024

You need to subtract the two cutouts on each side, so the side of the box would be 24 - 2x. X would then become the height of the box once folded.

The volume is found by multiplying the 4 dimensions:

The volume becomes x (24-2x) ^2

Now take the derivative of the formula to solve for x

Dv/dx = (24-2x) ^2 + x^2 (24-2x) * -2

= 24-2x (24-2x-4x) = 0

X = 12 or 4

24-2x = 24-2 (4) = 24-8=16

Volume = 16 x 16 x 4 = 1024 cubic inches.