water in a glass [now fully explained]

[Acidic]

This is the question I got at my York Uni interview, he drew I diagram for me and so I'll also try drawing and posting one.

here's the question:

and cylindrical glass is partially filled with water. The glass is tipped so that the water is a point at the open end (reaches but does not spil over) of the glass and just crosses the diameter of the bottom of the glass at the bottom.

Does the water in this glass fill 1/4, less than 1/4 or more than 1/4 of the glass and why?

OK, now my attempt at a diagram.

[alkatran]

First though: It's a quarter. Since it's going from half to 0.. half of half is 1/4.

Second thought: It's less than a quarter. The bottom part of the glass takes up less volume than the mid section. If you put the amount of water (and defied gravity) to fill it up to half way, you would need more (the ending point is wider...).

I'd go with less than a quarter, just from that logic.

[alkatran]

Here's a picture of what I mean:

[Acidic]

OK, I'll tell you it's tell than a quarter here was my initial logic (quite similair to yours I think).

If you reflected the shape of water onto the top of the glass, you should be left with a slice, one end the entire top of the glass, the other end a line, the diameter.

If you then cut this slice in 1/2, one end will cross at the diameter (same as the original water, but on other side), and the other side of the slice will go onto a line (whereas the original slice only goes to a point). This shows that the original slice is less that 1/4. This is correct, but he told me to provide better proof than that. I'll now try drawing what I've just explained.

[Acidic]

like my artistic skills?

[azteched]

what subject was this for?

It doesn't matter whether you get the questions right or not to be honest at nearly all uni interviews.

[Acidic]

it was for software systems engineering at York Uni.

I know it doesn't really matter, but it looks good.

I won't be going there anyways.

[fundu]

You are right. It is less than 1/4.

We can use integration or simple geometry to prove it.

I will try to put a sample solution on web tomorrow.

[Acidic]

yay, integration. That's how did it in the end.

[fundu]

There is another very simple and easy way to prove this. I mean to prove that it is less than a quarter.

Lets start from the last figure from Acidic in which he proved that the water is in shape of a half cone and took its projection.

So we can see that water forms a cone that is cut into half. Thus its volume will be half the volume of a cone.

Volume of cone is 1/3 * pi * r^2 * h

So volume of water here will be 1/2 * (1/3 * pi * r^2 * h) or in short 1/6 * pi * r^2 * h

We all know that volume of cylinder is pi * r^2 * h

Thus it is 1/6 times the full volume which is less than a quarter.

Howz that?

[Acidic]

Originally posted by fundu
Howz that?

confusing, but most probably correct. Here's how I did it in the interview.

draw 2-D cross-sectional diagrams at various length down the glass.

at bottom of glass:
water is half way up

half way up glass:
water is les than quarter way up. I proved it was less that 1/4 way up by the diagram attached.

at top of glass:
no water.

Then draw a graph of length up tube on x axis and area of water on y axis
when x=0, y=.5
when x=.5, y=less than .5, lets say .3
when x=1, y=0

this becomes an exponential decay curve. Integrating from 0 to 1 give you the total volume in the glass (the sum of al the areas). I proved this was less than 1/4 by drawing a straight line from (0,0.5) to (1,0) and showing my curve was always below this line (this is also on the diagram).

One thing to note, I didn't have a calculator, nor was I given any formulae so I couln't do your way.

OK, making diagram for next post.

[Acidic]

I have a typ on the image, the left hand side cross section is supposed to be the area of water at the bottom of the glass. also the text under the circles refers to the middle circle.

[fundu]

You are absolutely right Acidic. The graph is correct and even your analogy. The integration method gives you exactly this area.

But I think my suggestion was to use simple geometry method. As it is proving the point in 1 line.

Just use the area of cone and cylinder.

I think your response was fine and even your method. I wish you success in your interview.

[Acidic]

thanks, but I've already been offered a place there. I'll turn them down though as I would prefer going to Brunel in London.

[wossname]

Gee whizz your smart.

Yessiree.

[Acidic]

well, not quite. They've offered me a place on the condition that I get 36 points, equivelant to AAB. I know I won't get this, so I don't put them down as my choice. Brunel only asked for 30 points, roughly BBB or BBC

[azteched]

I'm sure you could get AAB.

[Acidic]

nope I can't, I'm good at off syllabus stuff, but now what I need to be good at. Anways, this is turning into a Chit Chat/PM conversation. should stop now.

[MikkyThomeon]

Yup

As an example I studied electronic control systems and then the mathematical modelling and Laplace freaked me out. At the same time I became interested in programming... so here I am now!!! Should have stuck with the more complicated stuff sadly. Too many programmers around in the world now.