To model the number of fencing panels needed to surround a square-shaped area as a function of the area, we can start from the relationship between the area of a square and the perimeter of a square.
- Area of a Square: If \( A \) is the area of the square, then \( A = s^2 \), where \( s \) is the length of a side of the square.
- Perimeter of a Square: The perimeter \( P \) (or the number of fencing panels needed) of a square is given by \( P = 4s \).
Combining these two relationships, we can express the perimeter in terms of the area:
\[ s = \sqrt{A} \] \[ P = 4s = 4\sqrt{A} \]
To express the number of panels needed in a function format:
\[ p(x) = 4\sqrt{x} \]
where \( x \) represents the area. This means that for a given area \( x \), the number of fencing panels needed \( p(x) \) is equal to \( 4\sqrt{x} \).
Since this specific option doesn’t appear in the responses provided, let's analyze the responses given:
- \( p(x) = \sqrt{x} - 2 \)
- \( p(x) = 1.25\sqrt{x} \)
- \( p(x) = \sqrt{x} - 1 \)
Among these options, the only function that seems to be a scaling of the square root of \( x \) is option 2, which is \( p(x) = 1.25\sqrt{x} \). This function suggests a relationship where the number of panels increases with the area but is scaled down by a factor of \( 1.25 \).
Thus, given the possible responses, the best choice is:
The number of fencing panels needed as a function of area is modeled by the function \( p(x) = 1.25\sqrt{x} \).