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