Cylinder in sphere

[Greenbank]

A classic maths puzzle which I was reminded of last night.

What is the largest cylinder (by volume) that can be inscribed in a sphere of radius R. Your answer should be in terms of R.

I like it because you start of with two unknowns and remove both of them. (Not counting R as an unknown because it is invariant.)

[drew]

Originally Posted by Greenbank

A classic maths puzzle which I was reminded of last night.

What is the largest cylinder (by volume) that can be inscribed in a sphere of radius R. Your answer should be in terms of R.

I like it because you start of with two unknowns and remove both of them. (Not counting R as an unknown because it is invariant.)

This is very similar to the coin problem. Some would argue that you can inscribe a rectangle inside a circle and maximize the area of the rectangle, yielding diameter=height. The volume (assuming a unit sphere) of the cylinder would be pi/(sqrt(2))^2*2/sqrt(2) = pi/sqrt(2) cubic units (about 2.221).

For the real answer, you need to optimize the 3 dimensional body.

This can be done using a system of equations. The volume of the cylinder and the
pythagorean theorem:
---
V=pi*radius^2*height
radius^2+(height/2)^2=1
---
substituting radius^2 from the second equation into the first yeilds a function of one variable:
---
V = pi*[1-(height/2)^2]*height
---
Minimize the volume
---
V/pi = height - height^3/4
d(V/pi) = 1 - 3*height^2/4
---
Setting the derivative = 0 yields a height of 2/sqrt(3). The radius, therefore, will be sqrt(2/3),
yielding a cylinder volume of 4/[3*sqrt(3)] (about 2.418) That's if I did my algebra correctly. I did
most of it in my head.


Edit: spoilerized text doesn't appear to line wrap. Is anyone aware of this bug?

Edit2: and when spoilerized posts are edited, it removes vertical whitespace. I had to insert --- lines to space out the content.

Drew

[Greenbank]

Yup, for a unit sphere 2.418 is correct to 3dp.

The answer I got was pi*4*R[sup]3[/sup] / ( 3 * 3[sup]1/2[/sup] ) (in terms of R).

Rough working (as I'll probably get bored of typing this out...)

Cylinder has radius r and height h.

By pythagoras: R[sup]2[/sup] = r[sup]2[/sup]+(h/2)[sup]2[/sup]

So: r[sup]2[/sup] = R[sup]2[/sup]-(h/2)[sup]2[/sup]

Volume of cylinder = pi*r[sup]2[/sup]*h

replacing r[sup]2[/sup] with R[sup]2[/sup]-(h/2)[sup]2[/sup] we get:

Vcyl = pi*(R[sup]2[/sup]-(h/2)[sup]2[/sup])*h

= pi*R[sup]2[/sup]h - pi*h[sup]3[/sup]/4

d Vcyl / dh = pi*R[sup]2[/sup] - pi*3*h[sup]2[/sup]/4

The stationary point is when d Vcyl / dh = 0 which occurs when:

pi*R[sup]2[/sup] = pi*3*h[sup]2[/sup]/4

so: R[sup]2[/sup] = (3/4) * h[sup]2[/sup]

Going back to Vcyl = pi*(R[sup]2[/sup]-(h/2)[sup]2[/sup])h

Vcyl = pi*(R[sup]2[/sup]-(h[sup]2[/sup]/4))h

We replace h[sup]2[/sup] by (4/3)*R[sup]2[/sup] and h by 2*R/sqrt(3)

Vcyl = pi*(R[sup]2[/sup] - (4/3)*R[sup]2[/sup]/4)*2*R/sqrt(3)

= pi*(R[sup]2[/sup]-(1/3)R[sup]2[/sup])*2*R/sqrt(3)

= pi*(2/3)*R[sup]2[/sup]*R/sqrt(3)

= pi*4*R[sup]3[/sup] / (3*sqrt(3))

or

= pi*4*sqrt(3)*R[sup]3[/sup] / 9

[drew]

Originally Posted by Greenbank

Yup, for a unit sphere 2.418 is correct to 3dp.

The answer I got was pi*4*R[sup]3[/sup] / ( 3 * 3[sup]1/2[/sup] ) (in terms of R).

Rough working (as I'll probably get bored of typing this out...)

Cylinder has radius r and height h.

By pythagoras: R[sup]2[/sup] = r[sup]2[/sup]+(h/2)[sup]2[/sup]

So: r[sup]2[/sup] = R[sup]2[/sup]-(h/2)[sup]2[/sup]

Volume of cylinder = pi*r[sup]2[/sup]*h

replacing r[sup]2[/sup] with R[sup]2[/sup]-(h/2)[sup]2[/sup] we get:

Vcyl = pi*(R[sup]2[/sup]-(h/2)[sup]2[/sup])*h

= pi*R[sup]2[/sup]h - pi*h[sup]3[/sup]/4

d Vcyl / dh = pi*R[sup]2[/sup] - pi*3*h[sup]2[/sup]/4

The stationary point is when d Vcyl / dh = 0 which occurs when:

pi*R[sup]2[/sup] = pi*3*h[sup]2[/sup]/4

so: R[sup]2[/sup] = (3/4) * h[sup]2[/sup]

Going back to Vcyl = pi*(R[sup]2[/sup]-(h/2)[sup]2[/sup])h

Vcyl = pi*(R[sup]2[/sup]-(h[sup]2[/sup]/4))h

We replace h[sup]2[/sup] by (4/3)*R[sup]2[/sup] and h by 2*R/sqrt(3)

Vcyl = pi*(R[sup]2[/sup] - (4/3)*R[sup]2[/sup]/4)*2*R/sqrt(3)

= pi*(R[sup]2[/sup]-(1/3)R[sup]2[/sup])*2*R/sqrt(3)

= pi*(2/3)*R[sup]2[/sup]*R/sqrt(3)

= pi*4*R[sup]3[/sup] / (3*sqrt(3))

or

= pi*4*sqrt(3)*R[sup]3[/sup] / 9

Oh yes...just realized I accidently omitted a factor of pi in my final expression. And of course, invoking similarity principles you can multiply my normalized solution by R^3 to get the solution for arbitrary R.

Drew

[mfgoode]

Originally Posted by drew

Oh yes...just realized I accidently omitted a factor of pi in my final expression. And of course, invoking similarity principles you can multiply my normalized solution by R^3 to get the solution for arbitrary R.

Drew

An excellent problem Greenbank'
Drew: try using paper and pen next time!
THe correct answers are
height =1.155*R
diameter of cylinder =1.633R
Volume of cylinder = 4*pi*R^3/(3*sq.rt.3). I leave that to you to work out!
Ratio of volume of cyclinder to that of the sphere is 1 : sq.rt.3
fraction of lost material is about 0.42 of the total material
Now isn't that simpler than your long drawn out spoilers?
And what benefit is it to others by leaving out pi and confusing all?
Mally

P.S. You must be a genius to work all that out in your head. Correction -lightning calculator would be more like it- definitely not genius.

[alpertron]

mfgoode, please next time use spoilers, as explained in this thread: http://www.mersenneforum.org/showthread.php?t=1866 so people can work on the problem without looking to the actual result unless they want to do it.

[drew]

mally,

I don't know what your problem is with my post. Forgive me for omitting the 'pi'...it was an oversight. Nothing else. Otherwise, I arrived at the same answer you had.

Originally Posted by mfgoode

Drew: try using paper and pen next time!

Uh...why? I arrived at the right answer, except for the missing 'pi' in the final result. Consider it a typo. The same could have occured if I had written it down.

Now isn't that simpler than your long drawn out spoilers?

Well, you only posted the result, not the method. Last time I tried that, you dismissed it and called it 'circumlocution'.

Let me ask you. Did it take you long to type out my result and try to insult me at the same time?

Drew

[mfgoode]

Originally Posted by alpertron

mfgoode, please next time use spoilers, as explained in this thread: http://www.mersenneforum.org/showthread.php?t=1866 so people can work on the problem without looking to the actual result unless they want to do it.

Thank you Alpertron for the URL.

Having been in the category of a senior manager in a reputed airline, I follow the following age old adage:

‘Rules were made for fools to follow;
Wise men should use their discretion’

In this particular case I have used my discretion and not ‘spoilerised’ my answers’
The reason is that if you scroll down this thread-its 90% ‘spoilerised’ by both posters, in method and answers. There has been no attempt to spread knowledge, rather, than to hide it.
It has become a ‘one on one’ exchange and if, that was intended, they should have used P.M.s and not the public forum.
You will see that the answers I have given, that too, in decimal and not irrational fractions, are not misleading at all as they don’t disclose the method that I have used to solve the problem.

In such a case I am justified to give an answer without spoilers which to me are not preferable. As a matter of fact it is more of a hint, should a poster, try to solve it by whatever method he prefers.
Instead of reprimanding the posters you have made the choice to ‘rap me on the knuckles’
No Alpertron, with due respect to you, this is not playing fair.
You are barking up the wrong tree!
Mally

[xilman]

Originally Posted by mfgoode

The reason is that if you scroll down this thread-its 90% ‘spoilerised’ by both posters, in method and answers. There has been no attempt to spread knowledge, rather, than to hide it.

Drivel.

The information has been spread to those willing to undertake the entirely insignificant cost of selecting the text with their mouse. Just because you seem to be too lazy is not an adequate excuse to prevent others from being able to work on a problem on their own and without the solution in full sight.


Paul

[cheesehead]

Mally,

I agree with alpertron and Paul about the desireability of using spoilers in your 19 Jan 06 post.

The reason is that if you scroll down this thread-its 90% ‘spoilerised’ by both posters, in method and answers. There has been no attempt to spread knowledge, rather, than to hide it.

This is a "Puzzles" subforum. Most of the value of a puzzle is in the working-out of it. That's why puzzle answers are traditionally not presented where the participant could unintentionally see them easily. Puzzles are not intended to serve as textbook examples for instruction, any more than exams traditionally have their solutions handed out right alongside the test problems.

Textbook examples spread knowledge; puzzles and exams are not intended to do that -- at least not until the reader has had a proper opportunity to try deriving the solution without being shown one.

It has become a ‘one on one’ exchange and if, that was intended, they should have used P.M.s and not the public forum.

In this public forum, those who, like myself, arrive a later than others want to have an equal opportunity to engage in mental exercise. That some responders had already exchanged a few relatively-non-answer-revealing comments doesn't justify eliminating that opportunity before a longer time has elapsed.

You will see that the answers I have given, that too, in decimal and not irrational fractions, are not misleading at all as they don’t disclose the method that I have used to solve the problem.

But disclosing the answers spoils much of the entertainment for all later-comers! It's not a matter of misleading. Now that I've seen them, I can't have as much delight in discovering them for myself!

spoilers which to me are not preferable.

So, start your own no-spoiler-blocking-allowed puzzle subforum. :-)

[mfgoode]

Originally Posted by cheesehead

Mally,

I agree with alpertron and Paul about the desireability of using spoilers in your 19 Jan 06 post.
So, start your own no-spoiler-blocking-allowed puzzle subforum. :-)

Instead of your suggestion, may I request the powers that be- administrators and moderators to work out some rule when an answer should be 'spoilerised' and when it need not be. Whether a method should be and when not.
By all means when an answer is liable to give the game away it should be spoilerised. If the method does the same, so be it Amen!.

Formerly in a previous post of mine Wacky took the initiative and spoilerised my answer as he felt it was too revealing for those like your self in the West.
Fair enough! But he had the courtesy in informing me that such an action was taken on his behalf and I agreed fully with him.

May I remind you that mathematics grew mainly out of puzzles first. There is
no dearth of examples but I presume you are well up in your history of mathematics to agree that this is very true.

I would once again request the powers to be to take up this topic and settle it once and for all and give the moderators the power to do so if they think fit.
As per the adage I am a fool and I would like a rule!
Mally