To analyze the meaning of the x-intercept of the function \( A = -w^2 + 60w \), we first need to determine where the x-intercepts occur. The x-intercepts of a function are the values of \( w \) for which \( A = 0 \).
Setting the equation equal to zero:
\[ -w^2 + 60w = 0 \]
Factoring out a \( w \):
\[ w(-w + 60) = 0 \]
This gives us two solutions:
- \( w = 0 \)
- \( -w + 60 = 0 \) which simplifies to \( w = 60 \)
The x-intercepts indicate the widths at which the area of the dog run is zero. Therefore, the width of the dog run must be either 0 feet or 60 feet. This means that the valid width for the dog run should be between these intercepts, which implies a width constraint.
Given the options provided:
- The dog run must have a width between 0 and 30 feet. (Incorrect)
- The dog run must have a width between 0 and 120 feet. (Incorrect)
- The dog run must have a width between 0 and 60 feet. (Correct)
- The dog run must have a width between 0 and 900 feet. (Incorrect)
The correct response is: The dog run must have a width between 0 and 60 feet.