The equation provided for the area \( A \) of the dog run is:
\[ A = -w^2 + 60w \]
This equation is a quadratic function in the standard form \( A = ax^2 + bx + c \), where \( a = -1 \) and \( b = 60 \). Since the coefficient of \( w^2 \) (which is \( a \)) is negative, this parabola opens downward, indicating that it has a maximum point.
To find the maximum area, we can use the formula for the vertex of a quadratic function, which gives the value of \( w \) at the maximum (or minimum). The \( w \) value at the vertex can be found using:
\[ w = -\frac{b}{2a} \]
Plugging in the values:
\[ w = -\frac{60}{2(-1)} = \frac{60}{2} = 30 \]
So, the width \( w \) that maximizes the area is 30 feet.
Next, we can find the maximum area by substituting \( w = 30 \) back into the area equation:
\[ A = -30^2 + 60 \times 30 \]
Calculating this:
\[ A = -900 + 1800 = 900 \]
Thus, the maximum area of the dog run that Nora can create with her fencing material is:
\[ \boxed{900} \text{ square feet} \]