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

Analyze Function A(x): To find the width that will produce the maximum garden area, we need to analyze the function A(x) = -x(x - 80). This is a quadratic function in the form of A(x) = ax^2 + bx + c, where a = -1, b = 80, and c = 0. The maximum value of a quadratic function ax^2 + bx + c occurs at x = -b/(2a).Calculate Maximum Value: First, we calculate the value of x at which A(x) reaches its maximum by using the formula x = -b/(2a). Here, a = -1 and b = 80. x = -80/(2 * -1) = -80/(-2) = 40.Identify Maximum Width: The width that will produce the maximum garden area is 40 meters. This is because the vertex of the parabola represented by the quadratic function A(x) = -x(x - 80) occurs at x = 40.Verify Maximum Point: We can verify that x = 40 is indeed the width that maximizes the area by checking that the second derivative of A(x) is negative at x = 40, which would confirm that it is a maximum point. The second derivative of A(x) with respect to x is A''(x) = -2, which is always negative, indicating that the function is concave down and thus has a maximum point at the vertex.

Simon has $160$ meters of fencing to build a rectangular garden. $\newline$ The garden's area (in square meters) as a function of the garden's width $x$ (in meters) is modeled by $\newline$ $A(x)=-x(x-80)$ $\newline$ What width will produce the maximum garden area? $\newline$ $\square$ meters

Full solution

More problems from Ratio and Quadratic equation

Related topics