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
𝐴
=
−
𝑤
2
+
60
𝑤
where A
𝐴
is the area and w is the width. Using this equation, identify the maximum range or area the fence can have.(1 point)
square feet

Answer 1

To find the maximum area that the fence can enclose, we can analyze the quadratic equation given:

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

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

The maximum value of a quadratic function occurs at the vertex, which for the equation \( w = -\frac{b}{2a} \).

1. Calculate \( w \):

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

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

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

Thus, the maximum area that Nora can enclose with the fence is \( \boxed{900} \) square feet.