Find the height of liquid in a partially filled upsidedown cone knowing the full height, diameter/radius at top, and volume of liquid in the cone.

Question

I can't for the life of me figure out how to calculate the height of water in a partially filled cone if I know the cone's full height, radius at top, and volume of water in the cone.

d = 92 cm
h = 33 cm
v = 36.64590267 Litres
x = ?

Please help!

Answer 1

Draw the perpendicular to the base from the vertex of the cone. The radius at the top is $R=d/2$. The radius at $x$ is $r$. You have similar triangles, so you can find $r$ in terms of $x$, $h$, and $R$. Now write the volume of the liquid in terms of $x$. Can you take it from here?

EDIT

Similar triangles: $$\frac hR=\frac xr$$ So $$r=\frac{Rx}h$$ The volume of the liquid in the cone is $$V=\frac13\pi r^2x=\frac13\pi \frac{R^2}{h^2}x^3=\frac{\pi d^2}{12h^2}x^3$$ Therefore $$x=\sqrt[3]{\frac{12Vh^2}{\pi d^2}}$$