Full solution

Q. Marquise has

200

meters of fencing to build a rectangular garden.

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The garden's area (in square meters) as a function of the garden's width

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x

(in meters) is modeled by:

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A(x)=-x^{2}+100x

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What is the maximum area possible?

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square meters

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 $h = -\frac{b}{2a}$ , 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 $h = -\frac{b}{2a}$ , we get $h = -\frac{100}{2*(-1)} = -\frac{100}{-2} = 50$ .
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.