Cylinder, sphere and cone.

mfgoode · 2006-05-26, 07:19

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

alpertron · 2006-05-26, 12:55

The graph is:

$\small\hspace{10}\unitlength{.75} \picture(250,250){~(125,225){\circle(200,40)}{~(125,25){\circle(200,40)}{~(25,30){\line(0,200)}{~(225,25){\line(0,200)}~(125,125){\circle(200,200)}{~(125,225){\line(-100,-200)}{~(125,225){\line(100,-200)}}$

Greenbank · 2006-05-26, 13:37

Cylinder:Sphere:Cone = 3:2:1

alpertron · 2006-05-26, 13:49

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.

mfgoode · 2006-05-26, 17:20

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

mfgoode · 2006-05-28, 04:34

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

alpertron · 2006-05-29, 20:04

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?
$\small\hspace{10}\unitlength{.75} \picture(250,150){~(125,25){\circle(200,20)}{~(125,125){\circle(200,20)}{~(125,75){\circle(200,20)}{~(125,75){\circle(175,15)}{~(125,125){\circle(200,200)}{~(25,25){\line(0,100)}{~(225,25){\line(0,100)}}$

alpertron · 2006-05-30, 16:48

I finally found the way to draw the sphere cap and cylinder!!!

$\small\hspace{10}\unitlength{.75} \picture(400,150){~(200,25){\circle(350,30)}{~(200,125){\circle(350,30)}{~(200,125){\circle(320,19)}{~(200,200){\circle(350,350;204,336)}{~(25,25){\line(0,100)}{~(375,25){\line(0,100)}}$

mfgoode · 2006-05-31, 04:09

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

2006-05-26, 07:19	#1
mfgoode Bronze Medalist Jan 2004 Mumbai,India 2²·3³·19 Posts	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

2006-05-31, 04:09	#9
mfgoode Bronze Medalist Jan 2004 Mumbai,India 2052₁₀ Posts	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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2006-05-26, 12:55	#2
alpertron Aug 2002 Buenos Aires, Argentina 3×7×73 Posts	The graph is: $\small\hspace{10}\unitlength{.75} \picture(250,250){~(125,225){\circle(200,40)}{~(125,25){\circle(200,40)}{~(25,30){\line(0,200)}{~(225,25){\line(0,200)}~(125,125){\circle(200,200)}{~(125,225){\line(-100,-200)}{~(125,225){\line(100,-200)}}$

2006-05-26, 13:37	#3
Greenbank Jul 2005 2×193 Posts	Cylinder:Sphere:Cone = 3:2:1

2006-05-26, 13:49	#4
alpertron Aug 2002 Buenos Aires, Argentina 3×7×73 Posts	Since h = 2r: Cylinder: pir^2h = 2pir^3 Sphere: (4/3)pir^3 = (4/3)pir^3 Cone: (1/3) r^2h = (2/3)pi*r^3 So what Greenbank wrote above is correct.

2006-05-26, 17:20	#5
mfgoode Bronze Medalist Jan 2004 Mumbai,India 2²×3³×19 Posts	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

2006-05-30, 16:48	#8
alpertron Aug 2002 Buenos Aires, Argentina 3·7·73 Posts	I finally found the way to draw the sphere cap and cylinder!!! $\small\hspace{10}\unitlength{.75} \picture(400,150){~(200,25){\circle(350,30)}{~(200,125){\circle(350,30)}{~(200,125){\circle(320,19)}{~(200,200){\circle(350,350;204,336)}{~(25,25){\line(0,100)}{~(375,25){\line(0,100)}}$