To analyze the meaning of the x-intercept of the function \( A = -w^2 + 60w \), we need to understand what the x-intercept represents in this context. The x-intercept occurs where the area \( A \) equals zero.
Setting the area equation to zero:
\[ -w^2 + 60w = 0 \]
We can factor this equation:
\[ w(-w + 60) = 0 \]
This gives us two solutions:
- \( w = 0 \)
- \( -w + 60 = 0 \) which simplifies to \( w = 60 \)
The x-intercepts \( w = 0 \) and \( w = 60 \) indicate the widths at which the area of the dog run is zero.
- \( w = 0 \) means there is no width, thus no area.
- \( w = 60 \) means that the width is at its maximum in terms of producing zero area.
Given that the width of the dog run must be between these two x-intercepts, we can understand that the dog run must have a width between \( 0 \) feet and \( 60 \) feet for it to have a positive area.
Therefore, the correct analysis of the x-intercept is:
The dog run must have a width between 0 and 60 feet.