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 $9/running foot and the steel fencing costs $3/running foot, determine the dimensions of the enclosure that will cost the least to erect. Round your answers to two decimal places.
A) 7.07 ft by 42.43 ft
B) 8.61 ft by 40.89 ft
C) 5.06 ft by 44.44 ft
D) 7.24 ft by 42.26 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 $9/running foot and the steel fencing costs $3/running foot, determine the dimensions of the enclosure that will cost the least to erect. Round your answers to two decimal places. 
A) 7.07 ft by 42.43 ft 
B) 8.61 ft by 40.89 ft 
C) 5.06 ft by 44.44 ft 
D) 7.24 ft by 42.26 ft

Quizplus · Accepted Answer

Let x be the length of the pine board sides and y be the length of the steel fencing side. The total cost C is given by:
C = 9(2x) + 3y
The area of the rectangle is 11eaa8b1_d997_4ae6_b1b6_b16aa0aea7b1_TB6026_11, so we have:
xy = 11eaa8b1_d997_4ae6_b1b6_b16aa0aea7b1_TB6026_11 - 11eaa8b1_d997_4ae6_b1b6_b16aa0aea7b1_TB6026_11
Solving for y in terms of x, we get:
y = (11eaa8b1_d997_4ae6_b1b6_b16aa0aea7b1_TB6026_11 - x)/x
Substituting this into the equation for C, we get:
C = 18x + 3(11eaa8b1_d997_4ae6_b1b6_b16aa0aea7b1_TB6026_11 - x)/x
Simplifying and taking the derivative with respect to x, we get:
dC/dx = 18 - 33/(x^2)
Setting this equal to zero and solving for x, we get:
x = sqrt(11eaa8b1_d997_4ae6_b1b6_b16aa0aea7b1_TB6026_11/3) ≈ 4.24
Substituting this back into the equation for y, we get:
y ≈ 2.61
So the dimensions of the enclosure that will cost the least to erect are approximately 4.24 ft by 2(4.24) + 2.61 ft = 11eaa8b1_d997_4ae6_b1b6_b16aa0aea7b1_TB6026_11, which is closest to the answer choice (A) 7.07 ft by 42.43 ft.

The Management of UNICO Department Store Decides to Enclose an