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?

Answer:w = w    L = 2w    h = hVolume:     V = Lwh                  10 = (2w)(w)(h)                  10 = 2hw^2                    h = 5/w^2Cost:    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 mCost:  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 costStep-by-step explanation: