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

cheapest cost = $125.58

Step-by-step 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 = (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