Question

Nora is creating a dog run for her dog, Mr. Darcey. She has enough fencing material for a fence with a perimeter of 120 feet. This situation is modeled using the equation A=−w2+60w
where A
is the area and w is the width. Using this equation, identify the maximum range or area the fence can have.(1 point)

Answer 1

The equation given for the area \( A \) in terms of the width \( w \) is:

\[
A = -w^2 + 60w
\]

This is a quadratic equation in the standard form \( A = ax^2 + bx + c \), where \( a = -1 \), \( b = 60 \), and \( c = 0 \).

To find the maximum area, we need to determine the vertex of the parabola described by this equation. The formula for the \( w \)-coordinate of the vertex of a quadratic equation \( y = ax^2 + bx + c \) is:

\[
w = -\frac{b}{2a}
\]

Substituting the values of \( a \) and \( b \):

\[
w = -\frac{60}{2(-1)} = \frac{60}{2} = 30
\]

Now, we substitute \( w = 30 \) back into the area equation to find the maximum area \( A \):

\[
A = -w^2 + 60w
\]
\[
A = -(30)^2 + 60 \cdot 30
\]
\[
A = -900 + 1800
\]
\[
A = 900
\]

Thus, the maximum area the fence can enclose is **900 square feet**.