Suppose we want to build a rectangular storage container with open top whose volume is $$12 cubic meters. Assume that the cost of materials for the base is$$‍12 dollars per square meter, and the cost of materials for the sides is $$8 dollars per square meter. The height of the box is three times the width of the base. What's the least amount of money we can spend to build such a container?

w = w L = 2w h = h

Volume: V = Lwh

10 = (2w) (w) (h)

10 = 2hw^2

h = 5/w^2

Cost: C (w) = 10 (Lw) + 2[6 (hw) ] + 2[6 (hL) ])

= 10 (2w^2) + 2 (6 (hw)) + 2 (6 (h) (2w)

= 20w^2 + 2[6w (5/w^2) ] + 2[12w (5/w^2) ]

= 20w^2 + 60/w + 120/w

= 20 w^2 + 180w^ (-1)

C' (w) = 40w - 180w^ (-2)

Critical numbers:

(40w^3 - 180) / w^2 = 0

40w^3 - 180 = 0

40w^3 = 180

w^3 = 9/2

w = 1.65 m

L = 3.30 m

h = 1.84 m

Cost: C = 10 (Lw) + 2[6 (hw) ] + 2[6 (hL) ])

= 10 (3.30) (1.65) + 2[6 (1.84) (1.65) ] + 2[6 (1.84) (3.30) ])

= $165.75 cheapest cost