To find the maximum area that Nora can achieve for the dog run, we need to determine the maximum value of the quadratic function \(A = -w^2 + 60w\). This function is a downward-opening parabola, and the maximum area occurs at its vertex.
The vertex of a quadratic equation in the form \(A = aw^2 + bw + c\) can be found using the formula \(w = -\frac{b}{2a}\).
In this case, the coefficients are:
- \(a = -1\)
- \(b = 60\)
Now, we can substitute these values into the vertex formula:
\[ w = -\frac{60}{2(-1)} = -\frac{60}{-2} = 30 \]
Next, we substitute \(w = 30\) back into the area equation to find the maximum area:
\[ A = -w^2 + 60w \] \[ A = -(30)^2 + 60(30) \] \[ A = -900 + 1800 \] \[ A = 900 \]
Therefore, the maximum area that Nora can achieve for Mr. Darcey's dog run is \( \boxed{900} \) square feet.