I have an interesting and challenging take on the old related rates problem involving a cone being filled with water. Suppose a cone with radius 10 feet and height 24 feet is being filled at the rate of 20 ft^3/min. How fast is the water rising when the height of the water is 6 feet?." Only instead of the cone being apex down (which is easy using similar triangles), it is lying on its side.
I know when you slice a cone with a plane you get a parabola. Therefore, the 'slices' will be a parabola sliced by a horizontal plane. The trick is getting this into an equation I can differentiate implcitly. Using the plane z=ay+c and the cone
z^2=a^2(x^2+y^z), I managed to derive y=ax^2/(2c)+c/(2a).
Now, what?. Does anyone have any insight?. I really would like to see this through. By the way, this is an interesting problem posed by a friend, not homework.
Thanks for any help or advice.