A closed-top rectangular box is to have a square base and a surface area of 100 square centimeters. What dimensions will maximize the volume?

Question

A closed-top rectangular box is to have a square base and a surface area of 100 square centimeters. What dimensions will maximize the volume?

Answer 1

Let x be the length of each side of the base. Let h be the heigh of the box. (x, h > 0)

The surface area is A = 2x² + 4hx = 100. So h = (50 - x²)/ 2x.

The volume of the box is V = x²h = x²(50 - x²)/ 2x = x(50 - x²)/2 = 25x - x³/2.

V'(x) = 25 - 3x²/2.

V'(x) = 0 when x = (5√6)/3 (since x > 0).

Since V'(x) > 0 for 0 < x < (5√6)/3 and V'(x) < 0 for x > (5√6)/3, V(x) is maximum when x = (5√6)/3.

So, h = (50 - ((5√6)/3)²)/(2(5√6)/3) = (5√6)/3 = x.

Thus, dimensions will maximize the volume are ((5√6)/3)·((5√6)/3)·((5√6)/3)