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
A
=
−
w
2
+
60
w
, where A
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 60 feet.
The dog run must have a width between 0 and 60 feet.

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 900 feet.

Answer 1

To analyze the meaning of the x-intercept of the function \( A = -w^2 + 60w \), we need to find the values of \( w \) (width) where the area \( A \) is equal to zero. The x-intercept occurs when \( A = 0 \).

Setting the equation to zero, we have:

\[
0 = -w^2 + 60w
\]

Factoring this equation, we get:

\[
0 = w(-w + 60)
\]

This gives us two solutions for \( w \):

1. \( w = 0 \)
2. \( -w + 60 = 0 \) which simplifies to \( w = 60 \)

These x-intercepts indicate the boundaries of width for the dog run. The area \( A \) represents a valid run only between these two intercepts. Therefore, the width \( w \) must be between 0 and 60 feet.

Thus, the best analysis of the meaning of the x-intercept is:

**The dog run must have a width between 0 and 60 feet.**