Lyon is attempting to build a fence for the largest possible rectangular play area at the local dog park. He can use up to 500 linear feet of fencing. He determines that if he makes a rectangular play area with a length of 103 or 147 feet of fencing, the play area would cover 15,504 square feet. To maximize the area, he can create a play area that is 15,625 square feet with a length of 125 feet. The relationship between the length, $ x $, and the area, $ y $, of the rectangular play area can be modeled with a quadratic equation.

Identify the vertex and another point from the context.

Vertex: $(125, 15,625)$

Point: $(103, 15,504)$ or $(147, 15,504)$

The vertex represents the maximum area Lyon can create with the fencing and is at (125, 15625). Another point from the context, which gives a different area, is at (103, 15504).

Explanation:

The given task is to identify the vertex and another point from the rectangular play area context with respect to the relationship between the length, x, and the area, y, using a quadratic equation.

The vertex of this quadratic, which represents the maximum area Lyon can create, is given as (125, 15625), indicating that with a length of 125 feet, the maximum area of 15,625 square feet is achieved.

Additionally, another point can be taken from the context where a length of 103 or 147 feet gives an area of 15,504 square feet.

Since we are looking for a single point, we can choose either, so let's use the length 103 feet, which yields a point as (103, 15504).