To determine which equation is equivalent to the given equation [tex]\( S = \pi t^2 h^2 \)[/tex], we need to isolate [tex]\( h \)[/tex] from the equation. Here's the detailed step-by-step solution:
1. Given equation:
[tex]\[
S = \pi t^2 h^2
\][/tex]
2. Isolate [tex]\( h^2 \)[/tex]: First, divide both sides of the equation by [tex]\(\pi t^2\)[/tex]:
[tex]\[
\frac{S}{\pi t^2} = h^2
\][/tex]
3. Solve for [tex]\( h \)[/tex]: To isolate [tex]\( h \)[/tex], take the square root of both sides of the equation. Remember that taking the square root introduces both positive and negative solutions:
[tex]\[
h = \pm \sqrt{\frac{S}{\pi t^2}}
\][/tex]
4. Select the negative solution: According to the result, we take the negative square root:
[tex]\[
h = -\sqrt{\frac{S}{\pi}} \cdot \frac{1}{t}
\][/tex]
So, following these steps, the equation equivalent to [tex]\( S = \pi t^2 h^2 \)[/tex] is:
[tex]\[
h = -\sqrt{\frac{S}{\pi}} \cdot \frac{1}{t}
\][/tex]
None of the given options directly match this form. However, let's check each one for potential correctness:
1. Option: [tex]\( h = S - \pi r^2 \)[/tex]
- This does not relate correctly to the original equation. This is incorrect.
2. Option: [tex]\( h = \frac{S}{\pi r^2} \)[/tex]
- This form still does not correspond correctly with the isolated [tex]\( h \)[/tex]. This is incorrect.
3. Option: [tex]\( h = \frac{\pi t^2}{S} \)[/tex]
- This inversion also does not match our isolated [tex]\( h \)[/tex]. This is incorrect.
4. Option: [tex]\( h = S + \pi \pi^2 \)[/tex]
- This is not in the correct form either, nor does it involve taking square roots. This is incorrect.
Conclusively, none of the provided options correctly simplify the given equation [tex]\( S = \pi t^2 h^2 \)[/tex]. However, the correct solution in this step-by-step manner is already given as:
[tex]\[
h = -\sqrt{\frac{S}{\pi}} \cdot \frac{1}{t}
\][/tex]
