Consider a square metal sheet that is 80 cm x 80 cm. Your task is to form an open box with the largest
possible volume by cutting equal square pieces from each corner of the sheet and then bending the flaps
upwards as in the illustration below. What are the dimensions of the box that yield the maximum possible
volume?

**ANSWER**:

*53 1/3 cm x 53 1/3 cm x 13 1/3 cm* which yields a volume of
37,925.9 cm

^{3}.

**EXPLANATION**: Note that calculus is required in one step of the solution to this puzzle.
Let

**x** be the height of the square that's cut out of each corner of the sheet. This also
represents the height of the open box.

Height = x

Length = 80 - 2x

Width = 80 - 2x

Volume = (80 - 2x)(80 - 2x)x

Volume = (6400 - 320x + 4x

^{2})x

Volume = 4x

^{3} - 320x

^{2} + 6400x

The next step requires the use of Calculus.

Maximizing:

0 = dV/dx = V'(x) = 12x

^{2} - 640x + 6400

0 = 3x

^{2} - 160x + 1600

0 = (3x - 40)(x-40)

x = 40/3 or x = 40. Since

**x = 40** results in a 0 value for width and height (80 - 2(40)),

**x = 40/3** yields the maximum volume. So the dimensions of the box are 53 1/3 cm x 53 1/3 cm
x 13 1/3 cm (or roughly 53.4 cm x 53.4 cm x 13.3 cm) with a volume of (40/3) x (80 - 2(40/3)) x
(80 - 2(40/3)) = 37,925.9 cm

^{3}.

Do you have a

suggestion for this puzzle (e.g. something that should
be mentioned/clarified in the question or solution, bug, typo, etc.)?