determine the dimensions of the enclosure that can be erected at minimum cost

Question

The management of the UNICO department store has decided to enclose an 813 ft2 area outside the building for displaying potted plants and flowers. One side will be formed by the external wall of the store, two sides will be constructed of pine boards, and the fourth side will be made of galvanized steel fencing. If the pine board fencing costs $6/running foot and the steel fencing costs $3/running foot, determine the dimensions of the enclosure that can be erected at minimum cost. (Round your answers to one decimal place.)

Answer 1

Let x = length of steel fence

     y = length of one side of pine boards

 

Given:  xy = 813.  So, y = 813/x

 

Minimize: C = cost of fence

 

                 = 3x + 6(2y) = 3x + 12y = 3x + 12(813/x)

 

C = 3x + 9756x^-1, x > 0

 

C' = 3 - 9756x^-2 = (3x²-9756)/x²

 

C' = 0 when 3x² = 9756

 

                   x² = 3252

 

                   x = √3252 ft ≈ 57.0 ft

 

Checking the sign of C', we see that C'<0 when 0 < x < √3252 and    

C'>0 when x > √3252.

 

So, the cost is minimized when x ≈ 57.0 and y = 813/x ≈ 14.3 ft