An rectangular area of 25 square meters, around an ancient historical site, is to be fenced to protect the artifacts. The sides with length x cost $ 34/ m and the other two sides cost $ 26 /m.
a. Write an expression for the cost as a function of x.
C(x) =
b. The value of x for minimum cost is x =
x w = 25 so w = 25/x
2 * 34* x + 2 * 26 * w = cost = c
c = 68 x + 52 w
c = 68 x + 52 (25/x)
c = 68 x + 1300/x
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dc/dx = 0 for min = 68 - 1300/x^2
x^2 = 1300/68
x = 4.37
a. To write an expression for the cost as a function of x, we need to consider the cost of each side of the rectangular area.
The two sides with length x each cost $34/m, so their combined cost would be 2x * $34.
The other two sides cost $26/m each, and since they are perpendicular to the first two sides, they will have a length of (25/x) meters. Therefore, their combined cost would be 2 * (25/x) * $26.
Adding these two costs together will give us the total cost as a function of x.
C(x) = (2x * $34) + (2 * (25/x) * $26)
b. To find the value of x for the minimum cost, we can take the derivative of the cost function C(x) with respect to x and set it equal to zero.
C'(x) = 2 * $34 - 2 * (25/x^2) * $26
Setting C'(x) = 0, we can solve for x algebraically:
2 * $34 - 2 * (25/x^2) * $26 = 0
Simplifying this equation will give us the value of x for the minimum cost.