Question

a rectangular box is to have a square base and a volume of 20 ft cubic if the material for the base costs 30 cent per square foot, the material for the sides cost 10 cent per sqaure foot, and the material for the top costs 20 cents per square foot, determine the dimentions of the box that can be construoted at minimum cost.

Answer 1

let the base be x by x
let the height be y
V = x^2y
20 = x^2y
y = 20/x^2

cost = 30x^2 + 4(10)xy + 20x^2
= 50x^2 + 40x(20/x^2)
= 50x^2 + 800/x

d(cost)/dx = 100x - 800/x^2
= 0 for min cost

100x^3 = 800
x = (8)^(1/3) = 2
y = 20/(2)^2 = 5

check my calculations.