A manufacturer needs to design an open box having a square base, and a total exterior surface area of 75 square inches. What dimensions will provide the largest possible volume?

Answer 1

dimensions for maximum Volume are 5 in by 5 in by 2 1/2 inches in height

surface area of an open box with square bottom = 75 in^2 = x^2 + 4xh

where x = side of the bottom and h= height of the box

x^2 + 4xh = 75

4xh = 75-x^2

h = 75/4x - x/4

volume = hx^2 = V = (x^2)(75/4x -x/4)= (75/4)x - x^3/4

take the derivative of V with respect to x, and set = 0

V'= dV/dx

= (75/4) -(3/4)x^2 = 0

multiply by 4

75 -3x^2 = 0

3x^2 = 75

x^2 = 75/3 = 25

x = 5 inches for each side of the box bottom (ignore the negative square root)

h = 75/4(5) - 5^2/4(5)= 75/20 -25/20 = 50/20 = 5/2 = 2.5 inches high

max Volume = x^2h = 25(2.5) = 62.5 cubic inches