The management of UNICO Department Store decides to enclose an area outside their building to display potted plants. The enclosed area will be a rectangle, one side of which is provided by the external walls 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 $15/running foot and the steel fencing costs $5/running foot, determine the dimensions of the enclosure that will cost the least to erect. Round your answers to two decimal places.
A) 14.31 ft by 84.68 ft
B) 12.13 ft by 86.86 ft
C) 14.14 ft by 84.85 ft
D) 15.68 ft by 83.31 ft

Question

The management of UNICO Department Store decides to enclose an  area outside their building to display potted plants. The enclosed area will be a rectangle, one side of which is provided by the external walls 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 $15/running foot and the steel fencing costs $5/running foot, determine the dimensions of the enclosure that will cost the least to erect. Round your answers to two decimal places. ​  ​
A) 14.31 ft by 84.68 ft 
B) 12.13 ft by 86.86 ft 
C) 14.14 ft by 84.85 ft 
D) 15.68 ft by 83.31 ft

Quizplus · Accepted Answer

Let the length of the pine board side be x, and the length of the steel side be y. Then the total cost C can be expressed as:
C = 15(2x) + 5y
C = 30x + 5y
   
The perimeter of the rectangle is given by:
P = 2x + y + 11eb651f_cae4_6808_b0a1_6b6b3119db62_TB8255_11
P = 2x + y + 66.55
   
We can solve for y in terms of x:
y = P - 2x - 66.55
   
Substituting into the equation for C, we get:
C = 30x + 5(P - 2x - 66.55)
C = -60x + 5P - 332.75
   
To minimize C, we take the derivative with respect to x and set it equal to zero:
dC/dx = -60 + 0 = 0
-60 + 5P = 0
P = 12.11
   
Substituting back into the equation for y, we get:
y = 12.11 - 2x - 66.55
y = -2x - 54.44
   
The dimensions of the enclosure are x by y, or x by -2x - 54.44. Since the length x must be positive, we choose the minimum value for C by setting the derivative equal to zero for the dimensions x = 7.07 ft and y = 12.72 ft, which gives us a rectangle with dimensions 14.14 ft by 84.85 ft. Therefore, the best choice is C.

The Management of UNICO Department Store Decides to Enclose an