10.  A rectangular storage container with an open top is to have a volume of 10 m^3.  The length of its base is twice the width.  Material for the base costs $10 per square meter.  Material for the sides costs $6 per square meter.  Find the cost of materials for the cheapest such 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