To solve for the variable [tex]\( h \)[/tex] in the equation [tex]\( v = \frac{1}{3} \pi r^2 h \)[/tex], follow these steps:
1. Start with the given equation:
[tex]\[
v = \frac{1}{3} \pi r^2 h
\][/tex]
2. Isolate [tex]\( h \)[/tex] on one side of the equation. To do this, we need to get rid of the [tex]\(\frac{1}{3} \pi r^2\)[/tex] term that is multiplied by [tex]\( h \)[/tex]. We can do this by dividing both sides of the equation by [tex]\(\frac{1}{3} \pi r^2\)[/tex].
3. First, represent [tex]\(\frac{1}{3} \pi r^2\)[/tex] in a more convenient form:
[tex]\[
\frac{1}{3} \pi r^2 = \frac{\pi r^2}{3}
\][/tex]
4. Now, divide both sides of the equation by [tex]\(\frac{\pi r^2}{3}\)[/tex]:
[tex]\[
h = \frac{v}{\frac{\pi r^2}{3}}
\][/tex]
5. Simplify the division by multiplying by the reciprocal of [tex]\(\frac{\pi r^2}{3}\)[/tex]:
[tex]\[
h = \frac{v \cdot 3}{\pi r^2}
\][/tex]
6. So, the expression for [tex]\( h \)[/tex] is:
[tex]\[
h = \frac{3v}{\pi r^2}
\][/tex]
By simplifying and substituting the given context, the solution for [tex]\( h \)[/tex] in terms of [tex]\( v \)[/tex] and [tex]\( r \)[/tex] is:
[tex]\[
h = 0.95492965855137 \frac{v}{r^2}
\][/tex]
This step-by-step process leads us to the desired solution.
