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. To analyze the meaning of the x-intercept: The dog run must have a width between _ and _ feet.

Answer 1

The function given for the area of the dog run is \( A = -w^2 + 60w \). This is a quadratic equation in the standard form \( A = ax^2 + bx + c \), where \( a = -1 \), \( b = 60 \), and \( c = 0 \).

To find the x-intercepts, we need to set \( A \) equal to 0 and solve for \( w \):

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

Factoring out \( w \):

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

This gives us two solutions:

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

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

Since \( w \) represents the width of the dog run and cannot be negative, the width must be strictly between these two values. Therefore, the dog run must have a width between:

\[
0 \text{ and } 60 \text{ feet}
\]

In conclusion, the width \( w \) must be in the range \( (0, 60) \) feet.