DFQ: "A vertical conical tank..."
I've spent probably 6 hours today trying to solve a basic differential equation, without success, and I'm about to go crazy.
"A vertical conical tank, point down, with a radius of 0.5 meters, and a height of 3 meters, is filled with water. How long does it take to drain the tank through a small hole in the bottom of the tank, when the water level has sunk by 1.5 meters after 45 minutes?"
The answer is supposed to be ~55 minutes.
I'm not gonna do the entire thing here, but here is general way of how I tried solving it:
1) Volume of a cone is =>
2) General solution to a "draining tank through a small hole" is
3) Rearranging and finding the derivative, getting
4) Integrating both sides =
5) Finding an expression for h(t):
6) Using t = 0 to find C, and then the information about the water level after 45 minutes, to find k.
7) Finding the time when h(t) = 0 (tank is empty). Standard second degree equation.
8) Getting the wrong answer. All but one time, I've gotten t = 2.56 hours, which is completely wrong.
Anyone able to see if there's anything wrong with that? Or am I just making some sort of fluke error while calculating? Anyone able to get it right (~55 minutes)?