Question

A landowner is planning to build a fenced-in rectangular patio behind his garage, using his garage as one of the "walls". he wants to maximize the area using 800n feet of fencing. The quadratic function A(x) = 80x - 2x² gives the area of the potion, where x is the width of one side. Find the maximum area of the patio.

Answer 1

The maximum area of the patio that the landowner can fence in using the quadratic function A(x) = 80x - 2x² is 800 square feet.

Step-by-step explanation:

To find the maximum area of the patio that can be fenced in, we need to consider the quadratic function given by A(x) = 80x - 2x², where x is the width of one side. To maximize the area, we can find the vertex of the quadratic function, since the vertex will give us the x-value that corresponds to the maximum y-value (in this case, the maximum area).

We know that the vertex of a quadratic function in the form of f(x) = ax² + bx + c is at x = -b/(2a). Here, a = -2 and b = 80. Substituting these values in gives x = -80/(2(-2)) = 20.

To find the maximum area, substitute x = 20 back into the original function: A(20) = 80(20) - 2(20)² = 1600 - 800 = 800 square feet.