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Volume of Spherical Cap

Date: 01/29/2002 at 14:09:45
From: Roger Staiger
Subject: Cone and sphere

There is a problem I cannot do; perhaps you can assist.

If a heavy sphere, whose diameter is 4 inches, be put into a conical 
glass full of water whose diameter is 5 and altitude 6 inches, how 
much water will run over? Ans: nearly 35/47 of a pint.

There were two kinds of pints in use back then: the "wine pint," which 
equalled 1/8 of a "wine gallon" or 28.875 (231/8) cubic inches, and 
the "ale pint," which equalled 1/8 of a "ale gallon" or 35.25 (282/8)
cubic inches.

Perhaps an answer in cubic inches is more useful since it avoids the 
units problem; but I did want to give you the answer.

Your assistance is appreciated.
Roger Staiger

Date: 01/29/2002 at 15:11:19
From: Doctor Peterson
Subject: Re: Cone and sphere

Hello again!

I think the text must have taught the formulas for a spherical cap; 
this one from the Dr. Math Geometric Formulas FAQ uses the volume 

    V = (Pi/6)(3r_1^2+h^2)h

where r_1 is the radius of the cross-section, and h is the distance 
from the center. Now consider the picture:

          |                |
          |                |
          |           ooooooooooo
          |        ooo     |     ooo
          |      oo       D|   r_1  oo F     E
     ---- +-----+----------+----------+-----+
      ^    \   o           |h      /   o   /
      |     \ o            |   /        o /
      |      \o           C+      r     o/
      |       o            |     \      o
      |        o           |           +
      |         o          |          o B
      |          oo        |        oo
      H           \ooo     |d    ooo/
      |            \  ooooooooooo  /
      |             \      |      /
      |              \     |     /
      |               \    |    /
      |                \   |   /
      |                 \  |  /
      |                  \ | /
      v                   \|/
     --------------------- +

Here R = 5/2, H = 6, r = 2. To find d, the height of the center of the 
sphere from the bottom of the cone, note similar triangles ABC and 

    d/r = AE/R = sqrt(R^2 + H^2)/R

Therefore d = r/R sqrt(R^2 + H^2), and h for our volume formula is 

    h = H - r/R sqrt(R^2 + H^2)

To find r_1, use the Pythagorean theorem on triangle CDF, giving

    r_1^2 = r^2 - h^2

I'll leave the rest to you: find the volume of the spherical cap BELOW 
the level of E, and that is the amount the spills. I got 0.7453, or 
about 35/47, of an ale pint.

- Doctor Peterson, The Math Forum   

Date: 01/29/2002 at 17:26:52
From: Roger Staiger
Subject: Geometry problem

Thank you again for solving my grandfather's grandfather's father's 
(his name was John E. Spare of Pottstown, PA) problem. I think there 
is another explaination for the spherical cap issue. I think it is 
less likely the text had an emphasis on spherical caps; and more 
likely that I never studied or used spherical cap. I don't ever 
remember studying spherical caps in my life. I don't ever think there 
was a need for spherical caps in my electrical engineering 

Thanks again. Your help was appreciated. Please write if I can ever 

Roger Staiger

Date: 01/29/2002 at 22:38:17
From: Doctor Peterson
Subject: Re: Geometry problem

Hi, Roger.

Did I say they emphasized spherical caps? I agree with you, just 
mentioning them at all makes his old text different from our experience. 
I don't think I learned about them before joining Dr. Math!

One reason I'm interested in your problems is that my family has a 
similar collection of schoolbooks and diaries from my great grandfather 
(and his father) going back only to the 1870's, but still of great 
historical and personal interest. To see how much more, in some ways, 
and less in others, the ordinary high school student learned back then 
gives an interesting perspective, doesn't it?

- Doctor Peterson, The Math Forum   
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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