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Simon has 160 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(x-80) What width will produce the maximum garden area? ◻ meters

Simon has 160160 meters of fencing to build a rectangular garden.\newlineThe garden's area (in square meters) as a function of the garden's width x x (in meters) is modeled by\newlineA(x)=−x(x−80) A(x)=-x(x-80) \newlineWhat width will produce the maximum garden area?\newline□ \square meters

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Q. Simon has 160160 meters of fencing to build a rectangular garden.\newlineThe garden's area (in square meters) as a function of the garden's width x x (in meters) is modeled by\newlineA(x)=−x(x−80) A(x)=-x(x-80) \newlineWhat width will produce the maximum garden area?\newline□ \square meters
  1. Analyze Function A(x)A(x): To find the width that will produce the maximum garden area, we need to analyze the function A(x)=−x(x−80)A(x) = -x(x - 80). This is a quadratic function in the form of A(x)=ax2+bx+cA(x) = ax^2 + bx + c, where a=−1a = -1, b=80b = 80, and c=0c = 0. The maximum value of a quadratic function ax2+bx+cax^2 + bx + c occurs at x=−b2ax = -\frac{b}{2a}.
  2. Calculate Maximum Value: First, we calculate the value of xx at which A(x)A(x) reaches its maximum by using the formula x=−b2ax = -\frac{b}{2a}. Here, a=−1a = -1 and b=80b = 80.x=−802×−1=−80−2=40x = -\frac{80}{2 \times -1} = -\frac{80}{-2} = 40.
  3. Identify Maximum Width: The width that will produce the maximum garden area is 4040 meters. This is because the vertex of the parabola represented by the quadratic function A(x)=−x(x−80)A(x) = -x(x - 80) occurs at x=40x = 40.
  4. Verify Maximum Point: We can verify that x=40x = 40 is indeed the width that maximizes the area by checking that the second derivative of A(x)A(x) is negative at x=40x = 40, which would confirm that it is a maximum point. The second derivative of A(x)A(x) with respect to xx is A′′(x)=−2A''(x) = -2, which is always negative, indicating that the function is concave down and thus has a maximum point at the vertex.

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