Quandaries
and Queries 

Hi I'm a grade 8 math teacher. There are 2 right cones, height is X. one of them is filled from the bottom up (round side) = x/2, when you invert the cone and add the same amount of liquid inside the second cone, what is the HEIGHT of the liquid? (thin side) you need 2 diagrams. 

Hi Vasuki, I drew a diagram of the cone filled to half the height and I labeled the radius of the base circle r.
The volume of a cone is
so the volume of this cone is
I am going to find the volume of liquid in the cone by subtracting the volume of the unfilled part of the cone. To do this I need to find the radius of the base of the unfilled part of the cone. This involves a use of similar triangles. In the diagram below triangles ABC and ADE are similar and hence
Thus DE = ^{r}/_{2} Thus the volume of liquid in the cone is
Now turn the cone over. Now I have labeled the height of the liquid h and the radius of the circle at the top of the liquid y. Thus the volume of the liquid is
But the volume of the liquid hasn't changed so
Again using similar triangles, this time from the diagram above
so
Substitute this value of y into
and solve for h. I got the cube root of ^{3}/_{4}, times x. Penny 

