Find the minimum cost of constructing the box.

Question

An open box with a rectangular base is to have a volume of 72 cubic feet. Its base has a length twice as long as its width. The material for the base will cost 50 cents per square foot and the sides will cost 25 cents per square foot. Find the minimum cost of constructing the box.

Answer 1

no answer

Answer 2

Volume = Lwh

 

   72 = (2w)w(h)

        = 2w^2h

 

  36 = w^2h

 

 

 

h = 36/w^2

 

 

The cost function in pennies is:

 

    (50) Lw    <---- base

       +

   2*(25) Lh <---  2 sides

 

      +

  2*(25) wh <----- front and back

 

=

 

50Lw + 50 Lh + 50 wh

 

50(2w)w+ 50(2w)h + 50 wh =

 

100w^2 + 100wh + 50 wh =

 

100w^2 + 150wh

 

100w^2 + 150w(36/w^2) =

 

100w^2 + 5400/w

 

= (100w^3 + 5400)/w

 

 

Differentiation by quotient rule:

 

 [w ( 300w^2) - (100w^3 + 5400) ] / w^2

 

 

 [ 200w^3 - 5400 ] / w^2

 

 

Setting equal to zero so as to minimize:

 

 200w^3 = 5400

 

 w^3 = 27

 

w=3

 

width is 3

length is 3*2 = 6

height is 36/3^2 = 36/9 = 4

 

The cost is (100w^3 + 5400)/w

 

 (100*27 + 5400) / 3 = 

 

(2700 + 5400)/3 =

 

 (8100)/3  = 

2700

 

which is $27 in pennies