So I am trying to work out a formula for the diameter of a partially filled cone with a non-zero diameter. For example, if I were filling a cup with water and I know the diameter of the bottom of the cup and either the volume of liquid I am pouring or the height of the liquid once it is poured, how can I calculate the diameter at the top of the water line?
I have been trying to use the formulas from here but I don't seem to be able to work out the formulas properly. Any help is greatly appreciated!!
edit: Here is an image to help with what I am trying to accomplish. I have D(bottom) and h and the slope of the wall, but I need D(top)
If you have the total volume and the height, you can use the formula for the cone: $$V=\frac{\pi h}{12}(D_{top}^2+D_{top}D_{bottom}+D_{bottom}^2)$$ You can rewrite this as: $$D_{top}^2+D_{top}D_{bottom}+D_{bottom}^2-\frac{12V}{\pi h}=0$$ Just apply the formula for quadratic, and choose the solution with $D_{top}\ge D_{bottom}$.
If you know the slope (let's call the angle $\alpha$), then $$\tan \alpha=\frac{R_{top}-R_{bottom}}{h}$$