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 :

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.