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The formula [tex][tex]$s=\sqrt{\frac{SA}{6}}$[/tex][/tex] gives the length of the side, [tex][tex]$s$[/tex][/tex], of a cube with a surface area, [tex][tex]$SA$[/tex][/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][tex]$\sqrt{30}-4\sqrt{5} \, m$[/tex][/tex]
B. [tex][tex]$\sqrt{30}-2\sqrt{5} \, m$[/tex][/tex]
C. [tex][tex]$\sqrt{10} \, m$[/tex][/tex]
D. [tex][tex]$2\sqrt{15} \, m$[/tex][/tex]

Sagot :

GinnyAnswer
To solve this problem, we need to find the side lengths of two cubes given their surface areas, and then determine how much longer the side of the first cube is compared to the second cube.

1. Find the side length of the first cube with a surface area of 180 square meters:

We use the formula for the side length [tex]\( s \)[/tex]:
[tex]\[ s = \sqrt{\frac{SA}{6}} \][/tex]
For a surface area ([tex]\(SA\)[/tex]) of 180 square meters:
[tex]\[ s_1 = \sqrt{\frac{180}{6}} = \sqrt{30} \][/tex]

2. Find the side length of the second cube with a surface area of 120 square meters:

Using the same formula, for a surface area ([tex]\(SA\)[/tex]) of 120 square meters:
[tex]\[ s_2 = \sqrt{\frac{120}{6}} = \sqrt{20} = 2 \sqrt{5} \][/tex]

3. Calculate the difference in side lengths between the two cubes:

The difference is given by:
[tex]\[ s_1 - s_2 = \sqrt{30} - 2\sqrt{5} \][/tex]

Given the side lengths and the difference calculated, the final step is to select the corresponding answer from the provided options:

[tex]\[ \sqrt{30} - 2\sqrt{5} \text{ m} \][/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.

So, the correct answer is:
[tex]\[ \sqrt{30} - 2\sqrt{5} \text{ m} \][/tex]
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