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Old 2006-05-26, 07:19   #1
mfgoode
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Question Cylinder, sphere and cone.


Here is an easy one provided you remember the formulae.

A cylinder whose height is the same as its diameter contains a sphere that exactly fits inside.

A cone also exactly fits inside when the sphere is removed.

What are the ratios of the respective volumes?

Alpertron how about using Tex to demonstrate?

Mally
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Old 2006-05-26, 12:55   #2
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The graph is:

\small\hspace{10}\unitlength{.75}    \picture(250,250){~(125,225){\circle(200,40)}{~(125,25){\circle(200,40)}{~(25,30){\line(0,200)}{~(225,25){\line(0,200)}~(125,125){\circle(200,200)}{~(125,225){\line(-100,-200)}{~(125,225){\line(100,-200)}}
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Old 2006-05-26, 13:37   #3
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Cylinder:Sphere:Cone = 3:2:1
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Old 2006-05-26, 13:49   #4
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Since h = 2r:

Cylinder: pi*r^2*h = 2*pi*r^3
Sphere: (4/3)*pi*r^3 = (4/3)*pi*r^3
Cone: (1/3) r^2*h = (2/3)*pi*r^3

So what Greenbank wrote above is correct.
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Old 2006-05-26, 17:20   #5
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Thumbs up


Excellent Greenbank and Alpertron for the sketch and answer.

Note that the area of a spherical cap of depth h is equal to a a ring of the cylinder of the same width h.

Mally
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Old 2006-05-28, 04:34   #6
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Smile Spherical cap in section

Quote:
Originally Posted by mfgoode

Excellent Greenbank and Alpertron for the sketch and answer.

Note that the area of a spherical cap of depth h is equal to a a ring of the cylinder of the same width h.

Mally

Alpertron, I dont want to make a habit of this. Please can you display your dexterity with TEX and cut off from the top of your sketch about 2 cms. so only that portion remains (The top 2 cms), to be able to visualise what I mean viz: that the area of the spherical cap is equal to the ring of the cyclinder of the same depth. You may or may not ,as you wish remove, that part of the cone which remains which ever is easier for you.
Thank you,
Mally
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Old 2006-05-29, 20:04   #7
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Quote:
Originally Posted by mfgoode

Alpertron, I dont want to make a habit of this. Please can you display your dexterity with TEX and cut off from the top of your sketch about 2 cms. so only that portion remains (The top 2 cms), to be able to visualise what I mean viz: that the area of the spherical cap is equal to the ring of the cyclinder of the same depth. You may or may not ,as you wish remove, that part of the cone which remains which ever is easier for you.
Thank you,
Mally
From the documentation it appears that only entire circles can be drawn.

What about this?
\small\hspace{10}\unitlength{.75}    \picture(250,150){~(125,25){\circle(200,20)}{~(125,125){\circle(200,20)}{~(125,75){\circle(200,20)}{~(125,75){\circle(175,15)}{~(125,125){\circle(200,200)}{~(25,25){\line(0,100)}{~(225,25){\line(0,100)}}
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Old 2006-05-30, 16:48   #8
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I finally found the way to draw the sphere cap and cylinder!!!

\small\hspace{10}\unitlength{.75}    \picture(400,150){~(200,25){\circle(350,30)}{~(200,125){\circle(350,30)}{~(200,125){\circle(320,19)}{~(200,200){\circle(350,350;204,336)}{~(25,25){\line(0,100)}{~(375,25){\line(0,100)}}
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Old 2006-05-31, 04:09   #9
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Thumbs up Cylinder and spherical cap.


Excellent Alpertron and thank you very much.
This is an improvement on your last sketch which also is very good.
I will be sending you my photos for my avatar as soon as I can get my scanner to do the job. How do I send it on to you? By PM, e-mail or what?
Mally
P.S. Dont bother really, but if you could have shaded the cap by dotted lines as in a section I think the result would be more outstanding

Last fiddled with by mfgoode on 2006-05-31 at 04:12
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