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

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 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 120 feet.
The dog run must have a width between 0 and 120 feet.

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

GPT-4o mini · Answer

To determine the meaning of the x-intercept in the function \( A = -w^2 + 60w \), we first need to find the x-intercepts by setting the area \( A \) to zero and solving for \( w \).

Setting \( A = 0 \):

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

Factoring the equation gives:

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

This leads to two solutions:

\( w = 0 \)
\( -w + 60 = 0 \) → \( w = 60 \)

The x-intercepts are \( w = 0 \) and \( w = 60 \).

This means that the width of the dog run can be 0 feet (indicating no width) or 60 feet (indicating that the area is maximized at that point). The width must therefore be between these two values, as negative widths do not make sense in this context.

Thus, the correct interpretation is:

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