The management of UNICO Department Store decides to enclose an 1440-ft 2 area ou

Question

The management of UNICO Department Store decides to enclose an 1440-ft2 area outside their building to display potted plants. The enclosed area will be a rectangle, one side of which is provided by the external wall of the store. Two sides of the enclosure will be made of pine board, and the fourth side will be made of galvanized steel fencing material. If the pine board fencing costs $8.00/running foot and the steel fencing costs $4.00/running foot, determine the dimensions of the enclosure that will cost the least to erect. (Give your answers correct to 1 decimal place.) in ft (steel side), ft (each pine side)

Explanation / Answer

Suppose each side of pine is P feet long.

So the cost of each side is 8P, and there are two pine sides, so the total cost of the pine is 16P.

Suppose the side of steel is S feet long. So the cost of the steel side is 4S.

The area of the rectange is PS = 1440,

so S = 1440/P

So the total fence cost is:

C = 16P+4S

C = 16P+4(1440/P)

C = 16P + 5760P^(-1)

In order to minimize C, let's find its derivative:

C' = 16 - 5760P^(-2)

Now let's find where the derivative is zero:

0 = 16 - 5760P^(-2)

16 = 5760/P^2

P^2 = 5760/16

P^2 = 360

P = sqrt(360)

P =60

S = 1440/P = 1440/60 = 24

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