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=−w2+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 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:

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