To analyze the meaning of the x-intercepts in the function \( A = -w^2 + 60w \), we need to find the values of \( w \) (width) for which the area \( A \) is equal to zero. The x-intercepts occur when the area of the dog run is zero, which means:
\[ A = -w^2 + 60w = 0 \]
We can factor this equation:
\[ w(-w + 60) = 0 \]
Setting each factor equal to zero gives us:
- \( w = 0 \)
- \( -w + 60 = 0 \) which simplifies to \( w = 60 \)
So, the x-intercepts are \( w = 0 \) and \( w = 60 \). This means that the area is zero when the width is 0 feet (indicating no width) or at the maximum width of 60 feet (where the area becomes negative for widths greater than 60 feet).
Given this information, we can analyze the provided answer options:
- The statement "the dog run must have a width between 0 and 900 feet" is incorrect, as 900 feet exceeds the maximum width of 60 feet.
- The statement "the dog run must have a width between 0 and 120 feet" is also incorrect, for the same reason.
- The statement "the dog run must have a width between 0 and 30 feet" is incorrect, as 30 feet is less than the valid maximum width of 60 feet.
- The statement "the dog run must have the width between 0 and 60 feet" is correct, as this statement accurately reflects the permissible widths based on the x-intercepts.
Therefore, the best analysis of the x-intercepts in this context is:
The dog run must have a width between 0 and 60 feet.