I do math problems in my head just to see what the answer REALLY could be....
....i'm so ashamed.
In class we've been covering the behavior of fluids. We were discussing density and buoyancy. I gave the class a fairly complex problem I knew they didn't have the tools to solve, then began walking them through it. Then I realized how very far beyond them it was and stopped. Then I realized the answer was fairly trivial and started working on a more difficult problem.
The easy problem was:
You have a cube of density ρc<ρf the density of a fluid. If the cube is of side length l, how far into the fluid will it sink?
It quickly becomes apparent that if you state that d is the length of the vertical edge below the surface of the water then d/l = ρc/ρf.
Now, a much more difficult problem is:
You have a right cone of height h, radius R, angle α, and density ρc in a fluid of density ρf, how far will it sink (base first) into the fluid?
You have to be able to find the roots of a cubic equation (it turns out to be surprisingly simple, actually. The equation, that is), but the answer is: d/h = (ρc/ρf - 1) ^ (1/3) + 1
You'll notice that this always gives an answer between 0 and 1, which is what you'd expect. Also, the closer the density of the cone is to the density of the fluid, the greater the depth to which the cone is submerged.
Then I tried to find out how far the cone would sink point first and made a mistake somewhere and came up with an answer such that the denser the cone, the less it's submerged. But I was way, way too tired to find my mistake.
