Question

Jacqueline has 100 feet of fencing to build a pen at her farm. She writes the function a=w(50-w) to represent the area of a pen with width w. Determine the vertex of the function and state what it means in the context of the scenario.

Answer 1

To find the vertex of the function a=w(50-w), rewrite the function in standard quadratic form and use the formula for finding the vertex. The vertex (-50/2(-1), f(-50/2(-1))) simplifies to (25, 625). In the context of the scenario, the vertex represents the maximum area of the pen that can be constructed with the given amount of fencing.

Step-by-step explanation:

To determine the vertex of the function a=w(50-w), we need to rewrite the function in the standard form of a quadratic equation: a=w(50-w) = 50w - w^2. The vertex of a quadratic function in the form of ax^2 + bx + c is given by (-b/2a, f(-b/2a)), where a, b, and c are the coefficients of the quadratic equation. In this case, the coefficient of w^2 is -1, the coefficient of w is 50, and there is no constant term, so c = 0.

The vertex is given by (-50/2(-1), f(-50/2(-1))). Simplifying, we find the vertex to be (25, 625). In the context of the scenario, the vertex (25, 625) represents the maximum possible area of the pen that can be constructed with the given amount of fencing. The width of the pen that gives the maximum area is 25 feet, and the maximum area that can be achieved is 625 square feet.