Question

The management of the UNICO department store has decided to enclose a 912 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 $5/running foot and the steel fencing costs $2/running foot, determine the dimensions of the enclosure that can be erected at minimum cost. (Round your answers to one decimal place.)

Answer 1

Answer:

the optimal dimensions of the rectangle in order to minimize cost are

19.1 ft x 47.74 ft

Step-by-step explanation:

Assuming that the area is rectangular shaped, then

Cost = cost of the pine board fencing * length of pine board fencing + cost of galvanized steel fencing * length of galvanized steel fencing

C = a*x + b*y

that is constrained by the area

Area=  A= x*y → y= A/x

replacing in C

C = a*x + b* A/x

the minimum cost is found when the derivative of the cost with respect to the length is 0 , then

dC/dx = a - b*A/x² = 0 → x = √[b/a*A]

replacing values

x = √[b/a*A] = √[($2/ft/$5/ft)*912 ft²] = 19.1 ft

then for y

y= A/x = 912 ft²/19.1 ft = 47.74 ft

then the optimal dimensions of the rectangle in order to minimize cost are

19.1 ft x 47.74 ft