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

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 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. 
    The dog run must have a width between 0 and 900 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.

GPT-4o mini · Answer

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 when \( A = 0 \). Setting the equation equal to zero gives us:

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

Factoring this equation:

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

This tells us that the x-intercepts are at \( w = 0 \) and \( w = 60 \).

Now, in the context of the dog run:

\( w = 0 \) means there is no width, and thus no dog run.
\( w = 60 \) means that when the width is 60 feet, the area of the run is also 0 because it implies that the length would be zero.

Therefore, the dog run must have a width between 0 and 60 feet, since these values represent the feasible widths where the area of the run will be non-negative and maximized for the area.

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

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