To find the maximum area that can be achieved with the given equation \( A = -w^2 + 60w \), we can recognize this as a quadratic equation in the standard form \( A = -w^2 + 60w \).
The maximum value of a quadratic equation \( ax^2 + bx + c \) can be found at the vertex, which occurs at \( w = -\frac{b}{2a} \). In this case, \( a = -1 \) and \( b = 60 \).
Calculating the width at which the maximum area occurs:
\[ w = -\frac{60}{2 \cdot -1} = \frac{60}{2} = 30 \]
Now that we have the width, we can substitute \( w = 30 \) back into the area equation to find the maximum area.
\[ A = -30^2 + 60 \cdot 30 \] \[ A = -900 + 1800 \] \[ A = 900 \text{ square feet} \]
Thus, the maximum area that the dog run can have is 900 square feet.