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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 is the maximum area possible? ◻ square 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 xx (in meters) is modeled by\newlineA(x)=−x(x−80)A(x)=-x(x-80)\newlineWhat is the maximum area possible?\newline□\square 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 xx (in meters) is modeled by\newlineA(x)=−x(x−80)A(x)=-x(x-80)\newlineWhat is the maximum area possible?\newline□\square square meters
  1. Perimeter Calculation: Simon has 160160 meters of fencing for a rectangular garden, which means the perimeter PP is 160160 meters. For a rectangle, P=2l+2wP = 2l + 2w, where ll is the length and ww is the width.
  2. Expressing Length in Terms of Width: Since the garden is rectangular, we can express the length in terms of the width as l=80−xl = 80 - x, because 2x+2(80−x)=1602x + 2(80 - x) = 160.
  3. Area Calculation: The area AA of the rectangle is given by A=l×wA = l \times w. Substituting l=80−xl = 80 - x, we get A(x)=x(80−x)A(x) = x(80 - x).
  4. Quadratic Function Analysis: The function A(x)=−x(x−80)A(x) = -x(x - 80) is a quadratic function that opens downwards (because of the negative sign), which means it has a maximum value at its vertex.
  5. Vertex Calculation: The vertex of a quadratic function in the form of f(x)=ax2+bx+cf(x) = ax^2 + bx + c is given by x=−b2ax = -\frac{b}{2a}. Here, a=−1a = -1 and b=80b = 80, so the x-coordinate of the vertex is x=−802∗(−1)=40x = -\frac{80}{2*(-1)} = 40.
  6. Maximum Area Calculation: Substitute x=40x = 40 into the area function to find the maximum area: A(40)=−40(40−80)=−40(−40)=1600A(40) = -40(40 - 80) = -40(-40) = 1600.

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