Question

Mr. Gordon wants to fence in his rectangular yard for his cat, Oscer. Based on his research he came up with the equation A=w (40 - w), where A is the area and w is the width of the space he fences in. What is the maximum area he can fence in? What will be the width for the maximum?

Answer 1

To find the maximum area Mr. Gordon can fence in for his cat's rectangular yard, we look at the equation A = w(40-w). The maximum area occurs at the vertex of the parabola, giving us a width (w) of 20 feet and a maximum area (A) of 400 square feet.

Step-by-step explanation:

Mr. Gordon is looking to determine the maximum area (A) he can fence in for his cat, Oscer, with a rectangular yard. The equation he has is A = w(40 - w), where w is the width. To find the maximum area, we need to analyze this quadratic equation, which is in the form of a parabola that opens downwards because the coefficient of w^2 is negative. This means the vertex of the parabola will give us the maximum area.

To find the vertex, we can complete the square or use the vertex formula w = -b/(2a). Here, a = -1 and b = 40, so the width for the maximum area is w = 40/(2*1) = 20 feet. Upon substituting w = 20 back into the equation, we get the maximum area A = 20(40 - 20) = 20(20) = 400 square feet.