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

Question

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 900 feet.
The dog run must have a width between 0 and 900 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 30 feet.
The dog run must have a width between 0 and 30 feet.
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Answer 1

To analyze the meaning of the x-intercept of the function \( A = -w^2 + 60w \), we first need to understand what the x-intercept represents. The x-intercept occurs when the area \( A \) is equal to zero. This means that the width \( w \) at the x-intercept is a dimension at which there is no enclosed area for the dog run.

To find the x-intercepts, we set the equation for the area equal to zero:

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

Factoring out \( w \):

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

This gives us two solutions:

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

The x-intercepts \( w = 0 \) and \( w = 60 \) indicate that when the width \( w \) is 0, the area is 0 (no run; no width), and when the width \( w \) is 60 feet, the area is also 0. This means that the maximum width allowed for the run must be between 0 and 60 feet to produce a positive area for the dog run.

Thus, the best answer among the options given is:

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