20060118, 17:11  #1 
Jul 2005
2×193 Posts 
Cylinder in sphere
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.) Last fiddled with by Greenbank on 20060118 at 17:13 
20060118, 17:55  #2  
Jun 2005
17E_{16} Posts 
Quote:
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 Last fiddled with by drew on 20060118 at 18:00 

20060118, 18:45  #3 
Jul 2005
602_{8} Posts 
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 
20060118, 19:05  #4  
Jun 2005
2×191 Posts 
Quote:
Drew Last fiddled with by drew on 20060118 at 19:06 

20060119, 16:38  #5  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
Cylinder in sphere
Quote:
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. 

20060119, 17:15  #6 
Aug 2002
Buenos Aires, Argentina
5FD_{16} Posts 
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.

20060120, 06:50  #7  
Jun 2005
2×191 Posts 
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. Quote:
Quote:
Let me ask you. Did it take you long to type out my result and try to insult me at the same time? Drew Last fiddled with by drew on 20060120 at 06:51 

20060120, 08:33  #8  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
Cyclinder in sphere.
Quote:
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 threadits 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 

20060120, 10:06  #9  
Bamboozled!
"𒉺𒌌𒇷𒆷𒀭"
May 2003
Down not across
2^{2}×2,971 Posts 
Quote:
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 

20060120, 14:02  #10  
"Richard B. Woods"
Aug 2002
Wisconsin USA
2^{2}×3×641 Posts 
Mally,
I agree with alpertron and Paul about the desireability of using spoilers in your 19 Jan 06 post. Quote:
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. Quote:
Quote:
Quote:
Last fiddled with by cheesehead on 20060120 at 14:04 

20060120, 15:12  #11  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
Quote:
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 

Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Phonograph Cylinder > Compact Disc  roger  Science & Technology  23  20081226 07:11 
Rotating cylinder  davieddy  Puzzles  7  20080409 01:12 
Triangles on a sphere  davieddy  Puzzles  15  20070406 20:16 
Cylinder, sphere and cone.  mfgoode  Puzzles  8  20060531 04:09 
Volume of a Sphere  Patrick123  Puzzles  19  20060220 15:00 