Express as a function of two variables.

Question

An open box is to contain a volume of 12 cubic meters. Given that the material for the sides of the box costs $2 per square meter and the material for the bottom costs $4 per square meter, express the total cost C of the box as a function of the length l and width w.

Answer 1

let the length be l and the width be w
let the height be h
lwh = 12
h = 12/(lw)

base is lw
sides are 2lh + 2lw

cost = 4lw + 2(2lh + 2hw)
= 4lw + 4lh + 4hw
= 4lw + h(4l+4w)
= 4lw + (12/(lw)(4l + 4w)
=4lw + 48/w + 48/l