Question

three hundred yards of fencing... to fence in a rectangular garden. the area of the garden is modeled by a quadratic function of the rectangle’s width, a(w). what does the second coordinate of the vertex of the quadratic function a(w) represent?
○ the minimum width that can be used for the fencing
○ the maximum area that can be enclosed by the fencing
○ the width that gives the maximum area
○ the length that gives the maximum area

Explanation:

The quadratic function \( A(w) \) models the area of the rectangular garden in terms of its width \( w \). For a quadratic function representing area (which opens downward, since area has a maximum), the vertex of the parabola represents the maximum point. The first coordinate of the vertex is the width \( w \) that gives the maximum area, and the second coordinate (the \( y \)-coordinate, or the value of \( A(w) \) at that vertex) represents the maximum area itself.

• The "minimum width" is not represented by the vertex (and the area function here is for maximizing, not minimizing width in this context).
• The "width that gives the maximum area" is the first coordinate of the vertex, not the second.
• The "length that gives the maximum area" is not represented by the second coordinate of the vertex of \( A(w) \) (which is a function of width, not length directly).

So the second coordinate of the vertex of \( A(w) \) is the maximum area that can be enclosed.

Answer:

the maximum area that can be enclosed by the fencing