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The formula [tex]\( S = \sqrt{\frac{SA}{6}} \)[/tex] gives the length of the side [tex]\( s \)[/tex] of a cube with a surface area [tex]\( SA \)[/tex]. How much longer is the side of a cube with a surface area of 180 square meters than a cube with a surface area of 120 square meters?

A. [tex]\( \sqrt{30} - 4\sqrt{5} \)[/tex] m
B. [tex]\( \sqrt{30} - 2\sqrt{5} \)[/tex] m
C. [tex]\( \sqrt{10} \)[/tex] m
D. [tex]\( 2\sqrt{15} \)[/tex] m



Answer :

GinnyAnswer
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]

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