Question

A rectangular storage container, complete with a special snap-on top, is to have a volume of 10m3 . The length of its base is twice the width. Material for the base and four sides costs $4 per square meter, but material for the special top costs $12 per square meter. Find, with justification, the cost of materials for the cheapest such container

Answer 1

Answer:

cheapest cost  = $125.58  

Explanation:

Given data:

w = w    L = 2w    h = h

Volume:     V = L×w×h

                 10 = (2w)(w)(h)

                 10 = 2hw^2  solving for h we have

                   $h = \frac{5}{w^2}$

Cost:    $C(w) = 10(Lw) + 2[4(hw)] + 2[4(hL)])$

                      $= 10(2w^2) + 2(4(hw)) + 2(4(h)(2w))$

                      $= 20w^2 + 2[4w(5/w^2)] + 2[8w(5/w^2)]$

                     $= 20w^2 + 20/w + 80/w$

                     $= 20 w^2 + 100w^{-1}$

            $C'(w) = 40w - 100w^{-2}$

Critical numbers:

          (40w^3 - 100)/w^2 = 0    multiply and divide by w^2

                      40w^3 -100 = 0

                              40w^3 = 100

                                  w^3 = 2.5

                                      w = 1.35 m

                                       L = 2.71 m

                                       h = 2.74 m

Cost:  C = 10(Lw) + 2[4(hw)] + 2[4(hL)])

              = 10(2.71)(1.35) + 2[4(2.74)(1.35)] + 2[4(2.74)(2.71)])

              = $125.58                      cheapest cost