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!
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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 $$x=\sqrt[3]{\frac{12Vh^2}{\pi d^2}}$$
Andrei
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I'm confused on how I'm to find r at the top of the liquid if I don't know the height of the liquid? – Nedder May 04 '21 at 16:31
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You write it in terms of $x$. Then when you write the volume of the liquid, you will have an expression that depends only on $x$. – Andrei May 04 '21 at 16:33
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I thank you for your help, but, math really isn't my thing. I'm a programmer trying to calculate this height using code. I would be eternally grateful if you could just provide me with the equation? I'm not a student cheating on school work :) I'm a 45 year old dude just trying to find an equation. – Nedder May 04 '21 at 18:14
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Thank you so much! I added this formula to my program and it works perfectly. You are a lifesaver! – Nedder May 04 '21 at 19:03