23.03.2023 - 16:11

A total of 2000 square feet is to be enclosed in two pens. The outside walls are to be constructed of brick, and the inner dividing wall is to be constructed of chain link. The brick wall costs $10 per linear foot and the chain-link costs $5 per linear fo

Question:

A total of 2000 square feet is to be enclosed in two pens. The outside walls are to be constructed of brick, and the inner dividing wall is to be constructed of chain link. The brick wall costs $10 per linear foot and the chain-link costs $5 per linear foot. Find the dimensions x and y that minimize the cost of construction.

Answers (1)
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    April 8, 2023 в 19:59

    To minimize the cost of construction, we need to determine the dimensions of the two pens such that the total cost of the brick and chain-link walls is minimized.

    Let x be the length of the rectangular pen and y be the width. Then the total area enclosed by the two pens is given by xy = 2000. We need to find the dimensions that minimize the cost of construction, which is given by:

    C(x,y) = 10(2x + 2y) + 5(x)

    where the first term represents the cost of the two brick walls (each of length x+y) and the second term represents the cost of the chain-link wall (of length x).

    We can simplify this expression to obtain:

    C(x,y) = 20x + 20y + 5x C(x,y) = 25x + 20y

    Now we can use the constraint xy = 2000 to eliminate one variable from the expression for C(x,y):

    C(x) = 25x + 4000/x

    We can find the minimum cost by taking the derivative of C(x) with respect to x and setting it equal to zero:

    dC/dx = 25 - 4000/x^2 = 0 x^2 = 4000/25 x = 40

    Substituting this value of x into the equation for C(x), we obtain:

    C(40) = 25(40) + 20(50) C(40) = 2000

    Therefore, the dimensions that minimize the cost of construction are x = 40 feet and y = 50 feet.

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