A box, with rectangular sides, base and top is to have a volume of 2 cubic feet. It has a square base. Express the surface area A of the box in terms of the width w of the base. If the material for the base and top costs 40 dollars/ft2 and that for the sides costs 50 dollars/ft2 express the total cost C as a function of the width.

Total cost F (w) = 160/w + 200 * √ (2w)

Step-by-step explanation:

Volume in cubic feet

V (box) = x*y*w and square base means x=y so V = 2 = x^ (2) * w

hence x^ (2) = 2/w (1)

Area of base and top in square feet, and cost in $

Area (t+b) = 2*x*y = 2x^ (2) C (1) = Cost of (base + top) C (1) = 40*2x^ (2)

C (1) = 80*x^ (2) and from eq. 1

C (1) = 80*2/w = 160/w

Area of sides = 4 * x * w = 4*√ ((2/w)) * w

C (2) = Cost of sides. is: C (2) = 50*4*√ ((2/w)) * w C (2) = 200 * √2w

Total Cost = F (c) = 160/w + 200*√2w