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

Question

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

Bytelearn · Accepted Answer

Problem description: The problem involves finding the maximum value of a quadratic function, which is given in the form A(x) = -x(x - 80). This is a parabola that opens downwards because the coefficient of the x^2 term is negative. The maximum value of this function occurs at the vertex of the parabola.Finding the x-coordinate of the vertex: To find the x-coordinate of the vertex, we use the formula -b/(2a), where the quadratic function is in the form ax^2 + bx + c. In our function A(x) = -x(x - 80), a = -1 and b = 80.Calculating the x-coordinate: Plugging the values of a and b into the vertex formula, we get -80/(2 * -1) = -80/(-2) = 40. This means the x-coordinate of the vertex, which gives us the width that will produce the maximum garden area, is 40 meters.Verification of calculation: To ensure there is no math error, we can check our calculation: -80 divided by -2 indeed equals 40.

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