To make [tex]\( h \)[/tex] the subject of the given relation:
[tex]\[
V = \pi h^2 \frac{(r - h)}{3}
\][/tex]
Follow these steps:
1. Multiply both sides by 3 to eliminate the denominator:
[tex]\[
3V = \pi h^2 (r - h)
\][/tex]
2. Divide both sides by [tex]\( \pi \)[/tex]:
[tex]\[
\frac{3V}{\pi} = h^2 (r - h)
\][/tex]
3. Expand the right-hand side:
[tex]\[
\frac{3V}{\pi} = h^2 r - h^3
\][/tex]
4. Rearrange into a standard cubic equation form:
[tex]\[
h^3 - h^2 r + \frac{3V}{\pi} = 0
\][/tex]
Now, the above cubic equation:
[tex]\[
h^3 - h^2 r + \frac{3V}{\pi} = 0
\][/tex]
is what we need to solve for [tex]\( h \)[/tex].
The solutions to this cubic equation for [tex]\( h \)[/tex] are:
[tex]\[
h = -\frac{r^2}{3 \left( \sqrt[3]{81V / (2 \pi) - r^3 + \sqrt{ -4r^6 + \left( 81V/\pi - 2r^3 \right)^2} / 2} \right)} + \frac{r}{3} - \frac{\sqrt[3]{81V / (2\pi) - r^3 + \sqrt{ -4r^6 + \left( 81V/\pi - 2r^3 \right)^2 } / 2}}{3}
\][/tex]
and the two complex conjugate solutions:
[tex]\[
h = -\frac{r^2}{3 \left( -\frac{1}{2} - \frac{\sqrt{3}i}{2} \right) \sqrt[3]{81V / (2\pi) - r^3 + \sqrt{ -4r^6 + ( 81V/\pi - 2r^3 )^2 } / 2}} + \frac{r}{3} - \left( -\frac{1}{2} - \frac{\sqrt{3}i}{2} \right) \frac{\sqrt[3]{81V / (2\pi) - r^3 + \sqrt{ -4r^6 + ( 81V/\pi - 2r^3 )^2 } / 2}}{3}
\][/tex]
and
[tex]\[
h = -\frac{r^2}{3 \left( -\frac{1}{2} + \frac{\sqrt{3}i}{2} \right) \sqrt[3]{81V / (2\pi) - r^3 + \sqrt{ -4r^6 + ( 81V/\pi - 2r^3 )^2 } / 2}} + \frac{r}{3} - \left( -\frac{1}{2} + \frac{\sqrt{3}i}{2} \right) \frac{\sqrt[3]{81V / (2 \pi) - r^3 + \sqrt{ -4r^6 + ( 81V/\pi - 2r^3 )^2 } / 2}}{3}
\][/tex]
