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Shenelle has 100 meters of fencing to build a rectangular garden. The garden's area (in square meters) as a function of the garden's width x (in meters) is modeled by: A(x)=-(x-25)^(2)+625 What is the maximum area possible? square meters

Shenelle has 100100 meters of fencing to build a rectangular garden. The garden's area (in square meters) as a function of the garden's width xx (in meters) is modeled by:\newlineA(x)=−(x−25)2+625A(x)=-(x-25)^{2}+625\newlineWhat is the maximum area possible?\newlinesquare meters \text{square meters}

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Q. Shenelle has 100100 meters of fencing to build a rectangular garden. The garden's area (in square meters) as a function of the garden's width xx (in meters) is modeled by:\newlineA(x)=−(x−25)2+625A(x)=-(x-25)^{2}+625\newlineWhat is the maximum area possible?\newlinesquare meters \text{square meters}
  1. Given Area Function: We are given the area function A(x)=−(x−25)2+625A(x) = -(x - 25)^2 + 625, where xx is the width of the garden in meters. To find the maximum area, we need to understand that the given function is a downward-opening parabola because the coefficient of the x2x^2 term is negative. The vertex of this parabola will give us the maximum area.
  2. Understanding the Parabola: The vertex form of a parabola is given by A(x)=a(x−h)2+kA(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola. In our case, the vertex is at (h,k)=(25,625)(h, k) = (25, 625), since the function is already in vertex form.
  3. Finding the Maximum Area: The vertex (25,625)(25, 625) tells us that the maximum area occurs when the width xx is 2525 meters. The maximum area is the kk value of the vertex, which is 625625 square meters.
  4. Checking the Perimeter: To ensure there is no math error, we can check if the total perimeter of the rectangle is indeed 100100 meters when the width is 2525 meters. The length would also be 2525 meters since the maximum area occurs when the garden is a square. The perimeter would be 2(length+width)=2(25+25)=2(50)=1002(\text{length} + \text{width}) = 2(25 + 25) = 2(50) = 100 meters, which matches the amount of fencing available.

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