To determine which of the given equations is equivalent to [tex]\( S = \pi r^2 n \)[/tex], let's analyze and manipulate the given equation step-by-step.
Starting with the given equation:
[tex]\[ S = \pi r^2 n \][/tex]
We are interested in isolating [tex]\( n \)[/tex] to find an equivalent expression.
Step 1: Solve for [tex]\( n \)[/tex].
To isolate [tex]\( n \)[/tex], divide both sides of the equation by [tex]\( \pi r^2 \)[/tex]:
[tex]\[ n = \frac{S}{\pi r^2} \][/tex]
Now let's compare this result with the given choices:
1. [tex]\( h = S - \pi r^2 \)[/tex]
2. [tex]\( n = \frac{S}{\pi r^2} \)[/tex]
3. [tex]\( h = \frac{\pi r^2}{S} \)[/tex]
4. [tex]\( h = S + \pi r^2 \)[/tex]
From our rearrangement, we see that the equation [tex]\( n = \frac{S}{\pi r^2} \)[/tex] matches our derived equation from the given equation [tex]\( S = \pi r^2 n \)[/tex].
Thus, the correct answer is:
[tex]\[ n = \frac{S}{\pi r^2} \][/tex]
So, the equivalent equation is:
[tex]\[ \boxed{n = \frac{S}{\pi r^2}} \][/tex]
