To determine which of the given equations is equivalent to the equation [tex]\( S = \pi \pi^2 n \)[/tex], follow these steps:
1. Analyze the given equation:
The equation given is:
[tex]\[
S = \pi \pi^2 n
\][/tex]
2. Evaluate each option one by one:
### Option 1: [tex]\( h = S - \pi \pi^2 \)[/tex]
- This equation does not involve the variable [tex]\( n \)[/tex] at all.
- Attempting to solve for [tex]\( n \)[/tex] from [tex]\( S = \pi \pi^2 n \)[/tex] by manipulating this equation would not yield an expression similar to [tex]\( h = S - \pi \pi^2 \)[/tex].
Hence, this option does not match the form of the given equation.
### Option 2: [tex]\( n = \frac{S}{\pi \pi^2} \)[/tex]
- Rearrange the given equation [tex]\( S = \pi \pi^2 n \)[/tex] to solve for [tex]\( n \)[/tex]:
[tex]\[
S = \pi \pi^2 n
\][/tex]
[tex]\[
n = \frac{S}{\pi \pi^2}
\][/tex]
This matches the second option exactly. Therefore, this option is equivalent to the given equation.
### Option 3: [tex]\( h = \frac{\pi \pi^2}{S} \)[/tex]
- Again, this equation does not involve [tex]\( n \)[/tex].
- Furthermore, trying to rearrange [tex]\( S = \pi \pi^2 n \)[/tex] into this form does not result in any meaningful similarity.
Thus, this option is not equivalent to the given equation.
### Option 4: [tex]\( h = S + \pi r^2 \)[/tex]
- This equation introduces [tex]\( r \)[/tex], which is not present in the original equation.
- It also does not involve a multiplication or division that would correspond to isolating [tex]\( n \)[/tex] from [tex]\( S = \pi \pi^2 n \)[/tex].
Consequently, this option too does not match the given equation in any form.
3. Conclusion:
After analyzing all the given options, the only equation that is mathematically equivalent to the given equation [tex]\( S = \pi \pi^2 n \)[/tex] is:
[tex]\[
n = \frac{S}{\pi \pi^2}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{2}
\][/tex]
