Originally posted by sonhouse
At work we have a coffee machine. We use styrofoam cups. I noticed the liquid is delivered in equal volumes, three times. The bottom of the styrofoam cup is 3 cm wide. The first shot made a depth of 1 cm, the second shot made a depth of 0.8 Cm and the third made a depth of 0.64 Cm. What is the angle of the conical shaped cup?
I want to bump this interesting thread. The problem is clever, it is well defined, and it is realistic. No loopholes, very clean.
I don't solve the problem now, I just give a way of solving it.
We know the area of the inside of a circle. Let's form a function f(r) showing the area as a function to it's diameter.
A cup is a part of a cone where the tip is cut off. The volume of this cup is an integreation of f(x) from point pb centimeters from the cone's tip to point pt centimeters. pb < pt.
Now we know that there is 1 unit of volume if you integrate f(x) from pb to pb+1 cm (I1), 1 unit of volume from pb+1 to pb+1.8 cm (I2), and 1 unit from pb+1.8 to pb+2.44 (=pt) cm (I3)
Now we need to calculate pb by finding out for wich pb I1 = I2 = I3.
When we know pb, then we can construct a triangle to find the angle.
I think we can solve the problem only with two observations, I3 is not neccesary.
This is a way to solution, if I'm right, perhaps I'm not.