The management of the UNICO department store has decided to enclose an 870 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 $4/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.)
If the steel side is x, then
xy = 870
and the cost
c = 2x+4*2y
so, sub in x or y and find dc/dx or dc/dy
98
To determine the dimensions of the enclosure that can be erected at the minimum cost, we need to find the dimensions that minimize the cost of the fencing.
Let's assume that the length of the side formed by the external wall of the store is x feet. The other two sides will be constructed of pine boards, so their combined length will be 2x feet. The remaining side will be made of galvanized steel fencing.
To find the cost function, we need to calculate the cost of the pine boards and the steel fencing.
The cost of the pine board fencing is $4 per running foot, and since there are two sides made of pine boards with a combined length of 2x feet, the cost of the pine board fencing is 4 * 2x = 8x dollars.
The cost of the steel fencing is $2 per running foot, and since there is one side made of steel fencing with a length of x feet, the cost of the steel fencing is 2 * x = 2x dollars.
The total cost function is the sum of the cost of the pine board fencing and the cost of the steel fencing:
Cost(x) = 8x + 2x = 10x
To minimize the cost, we need to find the value of x that minimizes the cost function. This can be done by finding the derivative of the cost function with respect to x and setting it equal to 0:
dCost/dx = 10 = 0
Solving this equation gives x = 0.
However, we cannot have a side length of 0, so we need to consider the endpoints of the interval we are working with.
Since we are dealing with a real-life scenario, x cannot be negative. The maximum value of x can be calculated by dividing the total area of 870 ft^2 by x, the length of the side formed by the external wall of the store:
x * 2x = 870
2x^2 = 870
x^2 = 435
x ≈ 20.9
Therefore, the valid range for x is from 0 to approximately 20.9.
Now we need to evaluate the cost function at the endpoints and the critical point to find the minimum cost:
Cost(0) = 10(0) = 0
Cost(20.9) = 10(20.9) ≈ 209
Comparing the values, we see that the minimum cost occurs when x = 0. Therefore, the dimensions of the enclosure that can be erected at the minimum cost is a length of 0 feet along the external wall and a combined length of 0 feet for the two sides constructed of pine boards.