The management of the UNICO department store has decided to enclose a 917 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 $7/running foot and the steel fencing costs $5/running foot, determine the dimensions of the enclosure that can be erected at minimum cost. (Round your answers to one decimal place.)
wood side ft
steel side ft
This is pretty much the same as the one I answered as well. Do it the same way.
no
To determine the dimensions of the enclosure that can be erected at the minimum cost, we need to find the lengths of the wood and steel sides.
Let's assume the length of the steel side is represented by x.
Therefore, the length of the wood side (which is made up of two pine board sides) would be (917 - x - x) ft = (917 - 2x) ft.
The cost of the wood side would be $7 per running foot, so the cost of the wood side would be:
Cost of wood side = Length of wood side * Cost per running foot = (917 - 2x) * $7.
Similarly, the cost of the steel side would be $5 per running foot, so the cost of the steel side would be:
Cost of steel side = Length of steel side * Cost per running foot = x * $5.
To minimize the cost, we need to minimize the total cost, which is the sum of the cost of the wood side and the cost of the steel side:
Total cost = Cost of wood side + Cost of steel side.
Now, we can formulate the equation to minimize the total cost:
Total cost = (917 - 2x) * $7 + x * $5.
To find the value of x that minimizes the total cost, we can differentiate the total cost function with respect to x and set it equal to 0:
d(Total cost)/dx = -14x + 4579 + 5x = 0.
Simplifying the equation, we get:
-9x = -4579.
x = -4579 / -9 = 508.8 ft (rounded to one decimal place).
Therefore, the length of the steel side is approximately 508.8 ft.
Substituting this value back into the equation for the length of the wood side, we get:
Length of wood side = 917 - 2x = 917 - 2 * 508.8 = 917 - 1017.6 = -100.6 ft.
Since the length of a side cannot be negative, we ignore this negative value as extraneous.
Therefore, the dimensions of the enclosure that can be erected at the minimum cost are approximately:
Wood side: N/A ft.
Steel side: 508.8 ft.