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Answer :
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.
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.
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