To find the maximum area that the fence can enclose for Nora's dog run using the equation \( A = -w^2 + 60w \), we need to identify the vertex of the quadratic function represented by this equation.
The standard form of a quadratic equation is given by \( A = ax^2 + bx + c \). In this case:
- \( a = -1 \)
- \( b = 60 \)
- \( c = 0 \) (since there is no constant term)
The \( w \)-coordinate of the vertex of a parabola (which gives the maximum or minimum value) can be found using the formula:
\[ w = -\frac{b}{2a} \]
Substituting in our values for \( a \) and \( b \):
\[ w = -\frac{60}{2 \cdot -1} = \frac{60}{2} = 30 \]
Now that we have the width \( w = 30 \), we can substitute this value back into the area formula to find the maximum area:
\[ A = -w^2 + 60w \] \[ A = -30^2 + 60 \cdot 30 \] \[ A = -900 + 1800 \] \[ A = 900 \]
Thus, the maximum area that the fence can enclose is:
\[ \boxed{900} \text{ square feet} \]