Question

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 1

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

Explanation: