Sure, let's solve the given problem step-by-step:
We are given that the surface area [tex]\( S_A \)[/tex] of a cube is 480 square meters. Our objective is to compare the side length [tex]\( s \)[/tex] corresponding to this surface area with the given options.
For a cube, the surface area [tex]\( S_A \)[/tex] is given by:
[tex]\[ S_A = 6s^2 \][/tex]
We need to find the side length [tex]\( s \)[/tex] for which the surface area [tex]\( S_A \)[/tex] of the cube is 480 square meters. We solve for [tex]\( s \)[/tex] by rearranging the formula:
[tex]\[ s = \sqrt{\frac{S_A}{6}} \][/tex]
Substituting the given surface area [tex]\( S_A = 480 \)[/tex] into the equation:
[tex]\[ s = \sqrt{\frac{480}{6}} \][/tex]
Calculating the inside of the square root:
[tex]\[ \frac{480}{6} = 80 \][/tex]
So,
[tex]\[ s = \sqrt{80} = \sqrt{16 \cdot 5} = 4 \sqrt{5} \][/tex]
Next, we need to compare this side length with the given options:
1. [tex]\( \sqrt{5} \)[/tex]
2. [tex]\( \sqrt{35} \)[/tex]
3. [tex]\( \sqrt{210} \)[/tex]
4. [tex]\( 7 \sqrt{5} \)[/tex]
Let's evaluate each option numerically:
1. [tex]\( \sqrt{5} \approx 2.236 \)[/tex]
2. [tex]\( \sqrt{35} \approx 5.916 \)[/tex]
3. [tex]\( \sqrt{210} \approx 14.491 \)[/tex]
4. [tex]\( 7 \sqrt{5} \approx 7 \times 2.236 = 15.652 \)[/tex]
Now, let's compare [tex]\( 4 \sqrt{5} \)[/tex] to these numerical values:
[tex]\[ 4 \sqrt{5} \approx 4 \times 2.236 = 8.944 \][/tex]
Thus, the side length [tex]\( s \)[/tex] corresponding to the surface area of 480 square meters is approximately 8.944 meters. Comparing it to the given options, we can see that:
- [tex]\( \sqrt{5} \approx 2.236 \)[/tex] (smaller)
- [tex]\( \sqrt{35} \approx 5.916 \)[/tex] (smaller)
- [tex]\( \sqrt{210} \approx 14.491 \)[/tex] (larger)
- [tex]\( 7 \sqrt{5} \approx 15.652 \)[/tex] (larger)
Therefore, the side length [tex]\( 4 \sqrt{5} \)[/tex] meters corresponding to the given surface area of 480 square meters is closer to but still different than all the given options. The correct comparison would be in terms of evaluating whether the approximate numbers match any given options within some tolerance, but strictly speaking, [tex]\( 4 \sqrt{5} \approx 8.944 \)[/tex] does not exactly match any provided numerical equivalence.
