To find the maximum area that Nora can achieve for her dog run, we start with the equation given:
\[ A = -w^2 + 60w \]
This equation represents a quadratic function in the standard form \( A = ax^2 + bx + c \), where:
- \( a = -1 \) (which indicates that the parabola opens downwards),
- \( b = 60 \),
- \( c = 0 \).
The maximum value of a quadratic function occurs at the vertex. The formula for the width \( w \) at the vertex for a quadratic equation \( A = ax^2 + bx + c \) is given by:
\[ w = -\frac{b}{2a} \]
Substituting the values of \( a \) and \( b \):
\[ w = -\frac{60}{2 \times -1} = \frac{60}{2} = 30 \]
Now that we have the width \( w = 30 \) feet, we can substitute this back into the area equation to find the maximum area:
\[ A = -w^2 + 60w \] \[ A = -(30)^2 + 60 \times 30 \] \[ A = -900 + 1800 \] \[ A = 900 \]
Thus, the maximum area that the dog run can have is 900 square feet.