A student needs to make a square cardboard piece. The cardboard should have a perimeter equal to at least 92 inches. The function [tex]f(s)[/tex] relates the perimeter of a cardboard piece, in inches, to the length of its side in inches. Which of the following shows a reasonable domain for [tex]f(s)[/tex]?

A. [tex]23 \leq s \leq 46[/tex]
B. [tex]23 < s < 92[/tex]
C. [tex]s = 92[/tex]
D. [tex]s \geq 23[/tex]

To solve the problem of finding a reasonable domain for the function [tex]\( f(s) \)[/tex], which relates the perimeter of a square cardboard piece to the length of its side, we need to follow these steps:

1. Understand the formula for perimeter:
The perimeter [tex]\( P \)[/tex] of a square is calculated using the formula:
[tex]\[
P = 4s
\][/tex]
where [tex]\( s \)[/tex] is the length of one side of the square.

2. Set up the inequality for perimeter:
According to the problem, the perimeter should be at least 92 inches. Therefore, we set up the inequality:
[tex]\[
4s \geq 92
\][/tex]

3. Solve for the side length [tex]\( s \)[/tex]:
To find the minimum possible value of [tex]\( s \)[/tex], divide both sides of the inequality by 4:
[tex]\[
s \geq \frac{92}{4}
\][/tex]
[tex]\[
s \geq 23
\][/tex]

4. Determine the domain of the function [tex]\( f(s) \)[/tex]:
Since [tex]\( s \)[/tex] must be at least 23 inches to satisfy the perimeter condition, the reasonable domain for the function [tex]\( f(s) \)[/tex] is all values for [tex]\( s \)[/tex] that are 23 inches or greater.

5. Conclusion:
Therefore, a reasonable domain for [tex]\( f(s) \)[/tex] is [tex]\([23, \infty)\)[/tex]. This means the side length [tex]\( s \)[/tex] can be 23 inches or any larger value.

By following these steps, we can understand the parameters for which the given conditions for the square are met.