# The Golden Ball & the Oxford Professors

1. May 5, 2006

### eldrick

An excellent puzzle for you all - not overly hard, but a good tester :

We have a ball made of pure gold. 2 Oxford Professor had obtained it in illicit fashion during an archaeological dig in Peru. The 2 accomplices being born complexifiers, fell into a dispute about how they should divide up this valuable object.

One had a fancy to have a solid gold paperweight & as a ball is not much use for that purpose, decided that it must be a cylinder ; so he said,

" All I want is a cylinder from the ball & I can turn this up on the lathe in the laboratory. All the rest of it, the golden swarf, you shall have & you can sell it for a considerable sum. "

The 2nd Professor did some calculations & proved to his own satisfaction that any true cylinder from the sphere must contain less than half the volume of the sphere & so he agreed to the terms.

Was he wise ?

If the total weight of the of gold in the ball was 1 kg, what was the least weight of swarf his friend could make in turning a true cylinder from the golden ball ? "

2. May 5, 2006

### topside

Is it 0.63kg?

3. May 5, 2006

### eldrick

nope

it's a high-school maths problem ( but probably last year of high school )

4. May 5, 2006

### matt grime

It's a high school maths problem for 14 year olds at most. It's a simple maths problem for people who've done maths to A-level equivalent.

5. May 5, 2006

### eldrick

it involves some calculus ( to get a non-calculator answer ) - they didn't teach me that as a 14y old ( only when i was 17y & doing my A -level )

6. May 5, 2006

### Staff: Mentor

volume(cylinder) / volume(sphere) = 3 / 4*SQRT(2) = 0.53

So the mass of the carved off part is 0.47kg

Although I solved it as a square in a circle problem and extended that answer to 3-D. I need to think more about if that's fully valid....

Happy Friday folks!

(EDIT -- I just did it for the sphere, and get the same term in the differentiation, so same answer.)

Last edited: May 5, 2006
7. May 5, 2006

### Curious3141

Something's wrong. I get $$max(\frac{V_{cyl}}{V_{sphere}}) = \frac{1}{\sqrt{3}}$$ which approximates 0.58. Second part is 0.42.

8. May 5, 2006

### eldrick

correct !

can you give us the detailed workings - it's nice for the public record

9. May 5, 2006

### Curious3141

Really ? I self-learned Calc (up to basic Integral Calc, including Volumes of Revolution) "for fun" when I was 11 or so. School taught (bored) me with it again at 15-16 before O levels.

10. May 5, 2006

### Staff: Mentor

Yeah, what'd I do wrong, I wonder. Did you get

0 = d/dTheta ( sin(theta)cos(theta) ) ?

11. May 5, 2006

### Curious3141

Taking a section thru the middle of the sphere/cylinder it's obvious that to have any hope of maximising the cylinder's volume, you'll need to inscribe it symmetrically into the sphere (touching the inside).

Then, in the cross-section, it just becomes a simple rectangle within a circle problem. Let the height of the cyl. be h Then the radius r of the cylinder is $$r = \sqrt{R^2 - \frac{h^2}{4}}$$

$$V_{cyl} = \pi r^2h = \pi(R^2 - \frac{h^2}{4})(h)$$

Differentiate that wrt h, set it to zero and solve for h, then find V_cyl, divide by 4/3*pi*R^3.

12. May 5, 2006

### Curious3141

I did it trigonometrically as well, if you're using the same theta I'm using, there should be a square on the sine term.

More precisely, if $$\theta$$ is the angle subtended between a vertical line drawn from the center of the cylinder and a line drawn from the center of the cylinder to the point where the limiting disk of the cylinder touches the circumscribing sphere, then

$$V_{cyl} = \pi R^2\sin^2\theta (2R\cos\theta) = 2\pi R^3\sin\theta \sin{2\theta}$$

max V_cyl when $$\tan{\theta} = \sqrt{2}$$

Last edited: May 5, 2006
13. May 5, 2006

### eldrick

unfortunately for me, my library didn't stock maths primers then, i had to wait 'til i was 14y ole, won the local rotary club essay prize - tasted vol-a-vents for the 1st time, got a £15 book token - with which i bought my 1st maths book - boz & chaz

more unfortunately, i didn't get a chance to do a maths degree after - i ended up as a cardiologist

14. May 5, 2006

### eldrick

Could someone post the detailed workings from a - z , in "code" form - nice for the record !

Last edited: May 5, 2006
15. May 5, 2006

### Curious3141

You're a cardiologist ? I'm a Clinical Microbiologist ! Well, in training anyway. I've always felt I've missed my true calling by becoming a Physician.

16. May 5, 2006

### eldrick

I assume you mean you missed your true calling by not becoming a Physician , as a Microbiologist isn't considered as having missed their true calling if they pass their MBBS or MD & then specialise in Microbiology

A Clinical Microbiologist without a prior MBBS or MD , is not a Physician

17. May 5, 2006

### Curious3141

I have an MBBS. I am a Clinical Pathologist.

My greatest regret is not going to CalTech (I had admission and a scholarship). My parents wanted me to do Medicine locally.

I consider my true calling to be within the Physical Sciences, pure or applied. I regret doing Medicine, which I consider to be intellectually unstimulating and a waste of time. Pursuing a non-clinical, academically-oriented discipline as a postgraduate is a compromise, making the best of a bad deal.

Last edited: May 5, 2006
18. May 5, 2006

### eldrick

More intriguing !?

MBBS is a "English" degree

They don't offer it to the Far East Guyz unless they study in England for at least 3y post-clinical...

19. May 5, 2006

### Curious3141

Please educate yourself on what Medical Degrees most former British colonies offer in their Universities. Singapore is a former Brit colony, FYI.

20. May 6, 2006

### matt grime

Ah, you wanted the exact amount cut off, rather than verifying that the professor in question was correct. I think there is a non-calculus method for doing it, but can't remember it. It's a famous problem, and certainly appears in the books of either Martin Gardner or David Wells.