To determine which of the following equations is equivalent to [tex]\( S = \pi r^2 h \)[/tex], we need to isolate [tex]\( h \)[/tex] on one side of the equation.
Given equation:
[tex]\[ S = \pi r^2 h \][/tex]
We want to solve for [tex]\( h \)[/tex]. Here are the steps:
1. Divide both sides by [tex]\( \pi r^2 \)[/tex]:
[tex]\[
\frac{S}{\pi r^2} = \frac{\pi r^2 h}{\pi r^2}
\][/tex]
2. Simplify the right-hand side:
[tex]\[
\frac{S}{\pi r^2} = h
\][/tex]
Thus, the equation simplifies to:
[tex]\[ h = \frac{S}{\pi r^2} \][/tex]
Now we need to match this equation to one of the given options:
- Option 1: [tex]\( h = S - \pi r^2 \)[/tex]
- This is not correct because the term involving [tex]\(\pi r^2\)[/tex] is subtracted, not divided.
- Option 2: [tex]\( h = \frac{S}{\pi r^2} \)[/tex]
- This matches our derived equation.
- Option 3: [tex]\( h = \frac{\pi \pi^2}{S} \)[/tex]
- This does not match our derived equation and is incorrect.
- Option 4: [tex]\( h = S + \pi t^2 \)[/tex]
- This introduces an incorrect variable [tex]\( t \)[/tex] and is incorrect.
Therefore, the correct equivalent equation is:
[tex]\[ h = \frac{S}{\pi r^2} \][/tex]
Thus, the correct answer is [tex]\( \boxed{2} \)[/tex].
