Which of the following equations is equivalent to [tex]\( S = \pi r^2 n \)[/tex]?

A. [tex]\( h = S - \pi r^2 \)[/tex]

B. [tex]\( n = \frac{S}{\pi r^2} \)[/tex]

C. [tex]\( h = \frac{\pi r^2}{S} \)[/tex]

D. [tex]\( h = S + \pi r^2 \)[/tex]



Answer :

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]