To determine which of the given side lengths corresponds to a square with an area of 450 square inches, we follow these detailed steps:
1. Understanding the relationship:
The area [tex]\( A \)[/tex] of a square is given by the formula:
[tex]\[
A = s^2
\][/tex]
where [tex]\( s \)[/tex] is the side length of the square.
2. Objective:
We need to identify which of the given options for [tex]\( s \)[/tex] satisfies the equation [tex]\( A = 450 \)[/tex].
3. Evaluating the given side length options one by one:
Option 1: [tex]\( s = 15 \sqrt{2} \)[/tex]
[tex]\[
\text{Area} = (15 \sqrt{2})^2 = 15^2 \cdot (\sqrt{2})^2 = 225 \cdot 2 = 450
\][/tex]
The area computed here matches the given area of 450 square inches. So, [tex]\( 15 \sqrt{2} \)[/tex] is a possible side length.
Option 2: [tex]\( s = 15 \sqrt{3} \)[/tex]
[tex]\[
\text{Area} = (15 \sqrt{3})^2 = 15^2 \cdot (\sqrt{3})^2 = 225 \cdot 3 = 675
\][/tex]
The area computed here is 675 square inches, which does not match the given area of 450 square inches. So, this option is not correct.
Option 3: [tex]\( s = 112.5 \)[/tex]
[tex]\[
\text{Area} = 112.5^2 = 12656.25
\][/tex]
The area computed here is 12656.25 square inches, which does not match the given area of 450 square inches. So, this option is also not correct.
Option 4: [tex]\( s = 115.5 \)[/tex]
[tex]\[
\text{Area} = 115.5^2 = 13340.25
\][/tex]
The area computed here is 13340.25 square inches, which does not match the given area of 450 square inches. So, this option is incorrect too.
4. Conclusion:
Among the given options, the one side length that correctly yields an area of 450 square inches is:
[tex]\[
15 \sqrt{2} \text{ inches}
\][/tex]
So, the side length of the square that has an area of 450 square inches is [tex]\( 15 \sqrt{2} \)[/tex] inches.
