Certainly! Let's go through a detailed, step-by-step solution to determine how much longer each of the given cube sides is relative to the side length [tex]\(\sqrt{5} \, m\)[/tex].
### Step 1: Identify Cube Side Lengths
We are given the side lengths of four cubes:
1. [tex]\( \sqrt{5} \, m \)[/tex]
2. [tex]\( \sqrt{35} \, m \)[/tex]
3. [tex]\( \sqrt{210} \, m \)[/tex]
4. [tex]\( 7 \sqrt{5} \, m \)[/tex]
### Step 2: Determine Differences
We need to find the difference between each side length and [tex]\(\sqrt{5} \, m\)[/tex]:
1. Difference between [tex]\( \sqrt{35} \, m \)[/tex] and [tex]\( \sqrt{5} \, m \)[/tex]
2. Difference between [tex]\( \sqrt{210} \, m \)[/tex] and [tex]\( \sqrt{5} \, m \)[/tex]
3. Difference between [tex]\( 7 \sqrt{5} \, m \)[/tex] and [tex]\( \sqrt{5} \, m \)[/tex]
### Step 3: Compute Differences
Now, let's compute these differences:
1. Difference between [tex]\( \sqrt{35} \, m \)[/tex] and [tex]\( \sqrt{5} \, m \)[/tex]:
[tex]\[ \sqrt{35} - \sqrt{5} \approx 3.680 \, m \][/tex]
2. Difference between [tex]\( \sqrt{210} \, m \)[/tex] and [tex]\( \sqrt{5} \, m \)[/tex]:
[tex]\[ \sqrt{210} - \sqrt{5} \approx 12.255 \, m \][/tex]
3. Difference between [tex]\( 7 \sqrt{5} \, m \)[/tex] and [tex]\( \sqrt{5} \, m \)[/tex]:
[tex]\[ 7 \sqrt{5} - \sqrt{5} = 6 \sqrt{5} \approx 13.416 \, m \][/tex]
### Step 4: Provide Final Answers
Based on the computed differences:
- The side length of [tex]\( \sqrt{35} \, m \)[/tex] is approximately [tex]\( 3.680 \, m \)[/tex] longer than [tex]\( \sqrt{5} \, m \)[/tex].
- The side length of [tex]\( \sqrt{210} \, m \)[/tex] is approximately [tex]\( 12.255 \, m \)[/tex] longer than [tex]\( \sqrt{5} \, m \)[/tex].
- The side length of [tex]\( 7 \sqrt{5} \, m \)[/tex] is approximately [tex]\( 13.416 \, m \)[/tex] longer than [tex]\( \sqrt{5} \, m \)[/tex].
These are the lengths by which each side is longer than the initial side length of [tex]\( \sqrt{5} \, m \)[/tex].
