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

Question

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

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Given Area Function: We are given the area function A(x) = -(x - 25)² + 625, where x 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 x² term is negative. The vertex of this parabola will give us the maximum area.Understanding the Parabola: The vertex form of a parabola is given by A(x) = a(x - h)² + k, where (h, k) is the vertex of the parabola. In our case, the vertex is at (h, k) = (25, 625), since the function is already in vertex form.Finding the Maximum Area: The vertex (25, 625) tells us that the maximum area occurs when the width x is 25 meters. The maximum area is the k value of the vertex, which is 625 square meters.Checking the Perimeter: To ensure there is no math error, we can check if the total perimeter of the rectangle is indeed 100 meters when the width is 25 meters. The length would also be 25 meters since the maximum area occurs when the garden is a square. The perimeter would be 2(length + width) = 2(25 + 25) = 2(50) = 100 meters, which matches the amount of fencing available.

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: $\newline$ $A(x)=-(x-25)^{2}+625$ $\newline$ What is the maximum area possible? $\newline$ $\text{square meters}$

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