Late Friday Puzzler --Martini Splitting

Clarification

Alex Y

When I said a ring is etched every inch from the "base", I meant from the bottom of the liquid holding part of the glass, not from the part that rests on the table, which is one kind of "base"; or the opening of the glass, which is the "base" of the cone. Hopefully that ambiguity is no throwing anyone off.

Re: Late Friday Puzzler --Martini Splitting

Larry Barrett

I do not see a simple way to do this. I think the answer is that you should drink the martini down to a point on the side = 5"(.5*(1/3))/sqrt2 = 5"(.794)/sqrt2 = 2.81".

Since you have not specified the cone angle, I will assume that the angle is 90 degrees, and the height (h) of the cone (from the point at the bottom of the glass to the top where the martini level is at the 5" mark) is equal to the radius (r) of the cone at that point.

So with this assumption, V = 1/3 pi r*3 = 1/3 pi h*3. The triangle which, if rotated, forms the cone is a right triangle with one side = 5" and the other two sides = h, so 5*2 = 2 h*2, h=5/sqrt2.

We want to drink enough martini down to a new level (t) on the side of the glass so that the amount of martini remaining is equal to the amount you drank.

This smaller cone will have a height (k) and radius (a) proportional to the larger cone, so will have a volume V= 1/3 pi a*3 = 1/3 pi k*3. And we want this volume to be 1/2 the larger volume, so

1/3 pi k*3 = 1/2 (1/3 pi h*3) and k*3 = 1/2 h*3, k = (1/2)*(1/3) h.

The triangle which, if rotated, forms this smaller cone is also a right triangle with one side = t and the other two sides = k. So t*2 = 2k*2.

Substituting, t = k/sqrt2 = (1/2)*(1/3)h/sqrt2 = (5"/sqrt2)((1/2*(1/3))/sqrt2.

There must be an easier way to do this one.

Sorry  Not there yet

Alex Y

There is indeed an easier way, and it doesn't depend on the angle of the cone. But working with the 45* sloped version should get you to the right answer. I think you were right until the last line.

Then you need to figure out why this answer works for ANY cone, at which point the Aha! will hit you.

Re: Late Friday Puzzler --Martini Splitting

Henry Higginbotham

Emptying and tilting the glass until one side of the liquid is at the fifth line and the opposite point is halfway between the second and third seems like it should make an even split, but I haven't thought of a way to positively mark that halfway point.

I don't know...

Alex Y

Henry, I haven't been ignoring you; just trying to figure out how to determine whether or not you are right. It seems to me that this would be less than 1/2 of the drink left for the second person, but I might be looking at it incorrectly.

If you take a "slice" of the drink through the points that touch the base and the high and low points you mention, the remaining amount would indeed be 1/2 of the original. But as you take parallel slices further from the center, it seems that smaller portions of the slice will be liquid.

But I might be looking at this wrong--what is your rationale?

BTW, the answer I am looking for is an approximation, so "1/2 way between" is just fine. I was looking for a liquid height with the glass upright, but you may have found another answer.

Re: Sorry  Not there yet

Larry Barrett

I still do not see an easier way, but correcting my arithmetic I think the answer is to drink the martini from the 5" mark down to the 4" mark (actually, to the

5/(2*.3333) = 5/1.25992... = 3.968" mark.) I think this answer, if correct, will apply to any cone shaped glass. But I do not see why it is obvious that the answer is to drink 1" of the martini.

I like Henry's idea of tipping the glass, but still do not see an obvious answer. If the glass was a cylinder, tipping so that the martini level touched the top and bottom would be half.

Re: I don't know...

Henry Higginbotham

The rationale is that if you look at a plane passing through the vertical axis of the glass, it forms a triangle. If the line along one side of the glass is assumed to be the base, then the vertex at the fifth line is the height. A line from that vertex to the midpoint of the base gives two triangles with equal bases and equal height. Revolving each 360° results in a equal volume. I hope.

Correct!

Alex Y

And you could probably do something similar to what you did for the general case of a cone with any angle, just using a little trig (which will get cancelled out).

But there is a better way.

At the start, you have a cone with volume V. You want to reduce it to a cone with volume 1/2*V. However, that second cone is exactly the same shape as the first one. That means that every linear measurement of the smaller cone will be the cube root of 1/2 as big. (And every two-dimensional measure, such as surface area, will be 1/2 ^ (2/3) as big.)

Re: I don't know...

Alex Y

The intersecting plane you describe is the "slice" I was trying to describe. I agree that 1/2 of the area of that intersection with the cone is wet. But as you rotate the slicing plane, the drink will not rotate with it. and at 90* rotation, I think the wet area of that slice will be less than 1/2 of the total. But I can't quite visualize it. I may have to search for a conical container for a little empirical check.