To solve the problem of determining how much longer the side of a cube with a surface area of 180 square meters is compared to a cube with the surface area of 120 square meters, we follow these steps:
1. Identify the Formula:
The formula to calculate the side length [tex]\( s \)[/tex] of a cube given its surface area [tex]\( SA \)[/tex] is:
[tex]\[
s = \sqrt{\frac{SA}{6}}
\][/tex]
2. Calculate Side Length for 180 Square Meters:
Substitute [tex]\( SA = 180 \)[/tex] into the formula:
[tex]\[
s_{180} = \sqrt{\frac{180}{6}} = \sqrt{30}
\][/tex]
3. Calculate Side Length for 120 Square Meters:
Substitute [tex]\( SA = 120 \)[/tex] into the formula:
[tex]\[
s_{120} = \sqrt{\frac{120}{6}} = \sqrt{20}
\][/tex]
4. Find the Difference in Side Lengths:
Compute the difference between the two side lengths:
[tex]\[
\Delta s = s_{180} - s_{120} = \sqrt{30} - \sqrt{20}
\][/tex]
To further simplify, we know that:
[tex]\[
\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}
\][/tex]
Thus, the difference in side lengths is:
[tex]\[
\Delta s = \sqrt{30} - 2\sqrt{5}
\][/tex]
5. Verification of Result:
- Given choices:
[tex]\(\sqrt{30} - 4\sqrt{5} \, m\)[/tex]
[tex]\(\sqrt{30} - 2\sqrt{5} \, m\)[/tex]
[tex]\(\sqrt{10} \, m\)[/tex]
[tex]\(2\sqrt{15} \, m\)[/tex]
Comparing [tex]\(\sqrt{30} - 2\sqrt{5} \)[/tex] with the answer choices, we see that it matches the second option exactly.
Thus, the side of a cube with a surface area of 180 square meters is [tex]\( \sqrt{30} - 2\sqrt{5}\)[/tex] meters longer than the side of a cube with a surface area of 120 square meters.
