Question

Nora is creating a dog run in her backyard for her dog, Max. She has enough materials to create a run with a perimeter of 120 feet. This is represented by the function A=−w^2+60w , where A is the area of the run and w is the width. Which of the following best analyzes the meaning of the x-intercept?(1 point) Responses The dog run must have a width between 0 and 30 feet. The dog run must have a width between 0 and 30 feet. The dog run must have a width between 0 and 120 feet. The dog run must have a width between 0 and 120 feet. The dog run must have a width between 0 and 60 feet. The dog run must have a width between 0 and 60 feet. The dog run must have a width between 0 and 900 feet.

Answer 1

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:

1. \( w = 0 \)
2. \( -w + 60 = 0 \) 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.**