So, for a school project I was working on this problem where I had to find the relationship between the volume and height of water being filled in this frustum. Initially, I thought that it would be a linear relationship because of the formula $$ V = {\frac{1}{3}}πh({r^2}+rR+{R^2}) $$ But then realised that it would be wrong as R itself is changing, i.e., R is a function of h. Which means that the graph would be exponential like this
Now, this is all I need for the project but my question is can we determine the exact relationship between the volume and height of water? The key, ofcourse is to write R in terms of h and that is where I have been stuck.
I think there is one missing piece of information: what is the angle between the axis of the frustum and the lines at its surface? Alternatively, what is the height of the "missing cone" at the bottom of the frustum?
If I call $h_0$ the height of the missing cone, then $R = r\left(1 + \frac{h}{h_0}\right)$.
If I call $\alpha$ the angle between the axis and the surface, then $R = r + h\tan(\alpha)$.