Question

A rectangular box with a volume of 320 cubic units is to be constructed with a square base and top. The cost per square foot of the bottom is 15 cents, for the top 10 cents, and for the sides is 2.5 cents. Find the dimensions that minimize the cost of constructing the box

Answer 1

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

x^ y = volume
x^2 y = 320
y = 320/x^2

cost = 15x^2 + 10x^2 + 4(2.5)xy
= 25x^2 + 10x(320/x^2)
= 25x^2 + 3200/x
d(cost)/dx = 50x - 3200/x^2 = 0 for a min of cost
50x = 3200/x^2
x^3 = 64
x = 4
then y = 320/16 = 20

the box should have a base of 4 by 4 and a height of 20 units