A(x)=-(1)/(4)(x-25)^(2)+625 The area, A(x), of a rectangular enclosure that can be made from a limited amount of fencing is shown, where x is the length of one of the sides of the enclosure, measured in feet. What is the maximum area that can be enclosed in square feet?

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A(x)=-(1)/(4)(x-25)^(2)+625 The area,  A(x), of a rectangular enclosure that can be made from a limited amount of fencing is shown, where  x is the length of one of the sides of the enclosure, measured in feet. What is the maximum area that can be enclosed in square feet?

Bytelearn · Accepted Answer

Identify Parabola Form: We are given the function A(x) = -(1/4)(x - 25)^2 + 625, which represents the area of a rectangular enclosure. To find the maximum area, we need to find the vertex of the parabola represented by this function, as it is a downward-opening parabola (indicated by the negative coefficient of the squared term).Find Vertex Coordinates: The function is in the form of a quadratic equation A(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. In this case, h = 25 and k = 625. Since the coefficient a is negative, the vertex represents the maximum point of the parabola.Calculate Maximum Area: The maximum area that can be enclosed is the value of A(x) at the vertex. Since we have identified the vertex as (h, k) = (25, 625), the maximum area A(x) is k, which is 625 square feet.No Derivatives Needed: We do not need to take any derivatives or set the derivative equal to zero because the vertex form of the quadratic function directly gives us the maximum value for this downward-opening parabola.

$A(x)=-\frac{1}{4}(x-25)^{2}+625$ $\newline$ The area, $A(x)$ , of a rectangular enclosure that can be made from a limited amount of fencing is shown, where $x$ is the length of one of the sides of the enclosure, measured in feet. What is the maximum area that can be enclosed in square feet?

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