Marquise has 200 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^(2)+100 x What is the maximum area possible? square meters

Question

Marquise has 200 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^(2)+100 x What is the maximum area possible? square meters

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Area Function Analysis: The problem gives us the area function A(x) = -x^2 + 100x, where x is the width of the garden. To find the maximum area, we need to find the vertex of the parabola represented by this quadratic function, since the coefficient of x^2 is negative, indicating that the parabola opens downwards and thus has a maximum point.Vertex Form Calculation: The vertex form of a quadratic function is A(x) = a(x-h)^2 + k, where (h, k) is the vertex of the parabola. To find the vertex of the parabola given by A(x) = -x^2 + 100x, we can use the formula h = -b/(2a), where a is the coefficient of x^2 and b is the coefficient of x.Calculate Vertex Coordinates: In our function A(x) = -x^2 + 100x, a = -1 and b = 100. Plugging these values into the formula h = -b/(2a), we get h = -100/(2*(-1)) = -100/(-2) = 50.Find Maximum Area Width: The x-coordinate of the vertex, h, is the width that gives us the maximum area. Since h = 50, the width of the garden that gives the maximum area is 50 meters.Substitute Width into Area Function: To find the maximum area, which is the y-coordinate of the vertex, k, we substitute the x-coordinate of the vertex back into the original area function. So we calculate A(50) = -(50)^2 + 100(50).Calculate Maximum Area: Calculating A(50) gives us A(50) = -2500 + 5000 = 2500 square meters. This is the maximum area possible for the garden.Final Result: We have found the maximum area of the garden to be 2500 square meters when the width is 50 meters.

Marquise has $200$ meters of fencing to build a rectangular garden. $\newline$ The garden's area (in square meters) as a function of the garden's width $\newline$ $x$ (in meters) is modeled by: $\newline$ $A(x)=-x^{2}+100x$ $\newline$ What is the maximum area possible? $\newline$ square meters

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