Question

Nora is creating a dog run for her dog. She has enough fencing material for a fence with a perimeter of 120 feet, modeled by this equation A=-w^2+60w where A is the area and evidence the width. Identify the maximum range or area the fence can have

Answer 1

To find the maximum area that Nora can achieve with her dog run, we can use the given quadratic equation for the area:

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

This is a standard form of a quadratic equation \( A = ax^2 + bx + c \), where:
- \( a = -1 \) (which means the parabola opens downwards),
- \( b = 60 \),
- \( c = 0 \).

The maximum area for a quadratic equation occurs at the vertex, which can be found using the formula for the \( w \)-coordinate of the vertex:

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

Plugging in the values of \( a \) and \( b \):

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

Now that we have the width at which the area is maximized, we can substitute \( w = 30 \) back into the area equation to find the maximum area \( A \):

\[
A = -30^2 + 60 \times 30
\]
\[
A = -900 + 1800
\]
\[
A = 900 \text{ square feet}
\]

Thus, the maximum area that the dog run can have is:

\[
\boxed{900 \text{ square feet}}
\]