To solve the problem of finding how much longer the side of a cube with a surface area of 180 square meters is compared to the side of a cube with a surface area of 120 square meters, let's break it down step-by-step:
1. Formulas and Given Data:
- Given formula for the side length of a cube in terms of its surface area [tex]\( S = \sqrt{\frac{SA}{6}} \)[/tex].
- The surface area of the first cube ([tex]\( SA1 \)[/tex]) is 180 square meters.
- The surface area of the second cube ([tex]\( SA2 \)[/tex]) is 120 square meters.
2. Calculate the Side Lengths:
- For cube 1 with [tex]\( SA1 = 180 \)[/tex]:
[tex]\[
s1 = \sqrt{\frac{180}{6}} = \sqrt{30}
\][/tex]
- For cube 2 with [tex]\( SA2 = 120 \)[/tex]:
[tex]\[
s2 = \sqrt{\frac{120}{6}} = \sqrt{20} = 2\sqrt{5}
\][/tex]
3. Find the Difference in Side Lengths:
- The difference in side lengths is given by:
[tex]\[
\text{side difference} = s1 - s2 = \sqrt{30} - 2\sqrt{5}
\][/tex]
Therefore, the side of the cube with a surface area of 180 square meters is [tex]\(\sqrt{30} - 2\sqrt{5}\)[/tex] meters longer than the side of the cube with a surface area of 120 square meters. Hence, the correct answer is:
[tex]\[
\boxed{\sqrt{30} - 2\sqrt{5} \, \text{m}}
\][/tex]
