Question

An open top box with a square base is to be made so that it holds 3 cubic feet. Assuming the material on the base costs $3 per square foot and the material on the sides costs $2 per square foot, determine the size of the base that minimizes the total cost.

Answer 1

let be base be x ft by x ft
let the height be y ft

(x^2)(y) = 3
y = 3/x^2

Cost = 3(x^2) + 2(4xy)
= 3x^2 + 8x(3/x^2)
= 3x^2 + 24/x

d(cost)/dx = 6x - 24/x^2
= 0 for a min of cost

6x = 24/x^2
x^3 = 4
x = cuberoot(4)
= appr 1.6 ft

the base should be appr 1.6 ft by 1.6 ft

(the height is appr 1.19 ft)