Question

An open box with a square base is to have a volume of 12 ft^3

a) Find a function that models the surface area, S(x), of the box. Write your function S(x) in terms of x, the dimension of each side of the square base.
b) Find the box dimensions that minimize the amount of material used. Round your answer correctly to two decimal places

Answer 1

The function for the surface area of the box in terms of x is S(x) = x^2 + 48/x. The dimensions that minimize the surface area are x ≈ 2.88 ft for the square base, and h ≈ 1.44 ft for the height.

Step-by-step explanation:

To find a function that models the surface area, S(x), of a box with a square base and a volume of 12 ft3, we let x represent the dimension of each side of the square base. Since volume (V) = x2h, where h is the height of the box, we can express h in terms of x: h = 12/x2.

The surface area of the open box consists of the base and the four sides. Thus, S(x) = x2 + 4(x)(12/x2) = x2 + 48/x. To minimize S(x), we take the derivative of S(x) with respect to x, set it to zero, and solve for x. After finding the critical point, we use the second derivative test to confirm that it is a minimum.

Using calculus, the minimum surface area occurs at x ≈ 2.88 ft, and the corresponding height h ≈ 1.44 ft.