Originally posted by sonhouse
You have one of those plug in water pots, you try to fill it at the water fountain, which only shoots a stream of water up 10 Cm high and 5 Cm over. The pot is a cylinder 15 Cm high and 8 Cm diameter. How many liters max can you fill the pot just using that spout of water?
I assume the cylinder must be tilted in order to catch the water, otherwise this would be fairly trivial. So, I will assume that one point of the cylinder will touch the ground at the same elevation as the origin of the water spout.
The best configuration to catch the water will be when the bottom lip of the pot just intersects with the top of the parabola formed by the flowing water. If you move the tilted pot any closer, the water will just hit the side of the pot; if you move it any further away, you'll have to tilt the pot more which will make some of the water run out.
Now, I think I'll give an overview of my solution first as I'm trying to do the math on the fly and it's slightly more complicated than I thought (but I will post my calculations too):
1. Determine the angle the pot makes with the ground, and subsequently the angle the water makes with the side of the pot on the interior (use trigonometry)
2. Tilt the pot upright, centre it at the origin on the xy-plane, and see where the plane determined by the angle of the water with the pot intersects the xy-plane (use trigonometry)
3. After determining the equation of the plane z = f(x,y), and using the fact that the equation of the cylinder in x^2 + y^2 = 16, perform a double integral with the following form:
V = int((f(x,y)))dydx, with limits of integration y = -SQRT(16-x^2)...+SQRT(16-x^2), x = -4...+4
This will give you the maximum volume of water captured in the pot. Now off to calculate!