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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 side width will produce the maximum garden area? 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: A(x)=−(x−25)2+625A(x)=-(x-25)^{2}+625 What side width will produce the maximum garden area? meters \text{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: A(x)=−(x−25)2+625A(x)=-(x-25)^{2}+625 What side width will produce the maximum garden area? meters \text{meters}
  1. Understand the Problem: Understand the problem.\newlineWe are given a quadratic function A(x)A(x) that represents the area of a rectangular garden with respect to its width xx. We need to find the width that gives the maximum area.
  2. Analyze the Quadratic Function: Analyze the quadratic function. The function A(x)=−(x−25)2+625A(x) = -(x - 25)^2 + 625 is a downward opening parabola because the coefficient of the squared term is negative. The vertex of this parabola will give us the maximum area.
  3. Find the Vertex: Find the vertex of the parabola.\newlineThe vertex form of a parabola is given by y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex. For the function A(x)=−(x−25)2+625A(x) = -(x - 25)^2 + 625, the vertex is at (h,k)=(25,625)(h, k) = (25, 625).
  4. Determine the Maximum Area Width: Determine the width that gives the maximum area.\newlineThe width that gives the maximum area is the xx-coordinate of the vertex, which is 2525 meters.

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