Still no solution yet

I'm still thinking about the cone problem. I've now acquired a classic text on co-ordinate geometry that devotes substantial space to the conic sections. Brushing-up on this topic would probably be worthwhile in any case, but I'm hoping it will give me the apparatus to work on this problem.

More updates to follow in due course, I hope!

The mathematics problem from my previous post continues to occupy my thoughts. Whilst I have made some progress with a solution, I have come up against a seemingly insuperable barrier. One of the things I need to be able to determine is the ratio into which a section through a frustum of a cone divides the volume, as illustrated above.

I had approached the problem by imagining slicing the cone into discs. The section would then create cicrle segments whose areas could be integrated to obtain the volume. Unfortunately, the expression derived for the area in terms of the height is very hard to integrate. It seems such an obvious question to ask, I can't believe that a solution isn't out there somewhere!

Water in a glass

A scientist is someone who sees the world around them and wonders how it is like it is.

Even though I've been a scientist for most of my life, it still surprises me that so many interesting problems can arise from something as simple as a glass of carbonated water. Here are three that have occurred to me during the last week.

1. An economics problem: why is Sainsbury's own-brand carbonated water cheaper than their own-brand still water (by around 10%)? Surely, the carbonation process requires more input of energy and materials, so it should be more expensive.
2. A physics problem: when the bubbles rise they clearly accelerate, but is the acceleration constant, or is there a significant change of force with depth, resulting in non-negligible variable acceleration?
3. A mathematics problem: what is the relationship between the volume of water in the glass and the angle by which the glass can be tipped before the water spills? This would, of course, be the same regardless of the nature of the liquid in the glass, providing surface tension is ignored. I am assuming that the glass is a frustum of a right circular cone.
   

The economics question doesn't really interest me, although the answer might.

The physics problem could probably be answered with a bit of thought and some back-of-an envelope calculations and might make a useful diversion when stuck invigilating internal examinations next week.

It is the mathematics problem that has really gripped me. Despite appearing fairly straightforward initially, it turns out to be really quite complex, requiring a determination of volume of a conic section. There are 2 cases to consider:

a) a fairly full glass, where the liquid forms an eliptical section of the frustum;



b) a fairly empty glass, where the liquid forms a section of the frustrum in the form of a elipse segment (the base of the glass constituting the chord).



Finding the air/liquid contact area would be straight forward, but extending to the volume seems to be quite tricky. A volumes of revolution approach will not work as the volume section has no cylindrical symmetry. The internet doesn't seem to be very helpful, but then I might not be constructing a suitable search as the sections I'm interested in might have a "proper" name. My next stop will be my Euclid, although I don't hold out much hope.

Any suggestions would be most gratefully received.

Of course, being a scientist I could just be a bit empirical about this: I could go and get some data and then see if I can find a relationship. Maybe that should be my next stop ...

Halcyon days ...

... are here again.

There is a short period in a teacher's year when examination classes have departed for study leave and internal examinations and reports are not yet threatening, so that workload eases for just a couple of weeks. This calm is perhaps the best time of year - time to relax, reflect and generally reorganise one's life.

"Oh, come on," I hear you exclaim, "what about those long summer holidays?". Somehow, they never seem to be the treat that you think they'll be. There's always something to be doing, either domestic chores or the planning that you hope will ease the burden of the new academic year; somehow several weeks of "free" time are never so greatly enjoyed as a rare work-free weekend.

I'm hoping to finish The Daffodil Affair (Michael Innes) and Nature's Chemicals (Richard Firn) and there's a fern to plant out (once I've dug up an the remains of old ornamental current tree) - photos to follow.