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### Volume of a Partial Sphere

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Date: 01/07/98 at 18:50:46
From: Thomas Lee
Subject: Volume of a partial sphere

If I take a spherical container filled with fluid and then introduce a
bubble of gas into the container (which displaces the fluid so the
bubble's volume remains unchanged), what would be the surface of the
sphere which is in contact with the bubble (assuming the bottom
meniscus of the bubble is flat - i.e. that part which is not in
contact with the container) given the volume/radius of the sphere and
the volume of the bubble?

I am asking this because I will be giving a talk on the use of
intra-ocular gases in eye surgery.
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Date: 01/08/98 at 08:35:32
From: Doctor Jerry
Subject: Re: volume of a partial sphere redux

Hi Thomas Lee,

As I understand the problem, you have a sphere of radius a (this is
the container) into which is put a (small) bubble of gas, say of
radius b.  In my visualization, this small bubble rises to the top of
the container and deforms to something fitting against the sphere on
top and having a flat bottom. I take it that the physics of the
situation makes this at least approximately true.

You ask "what would be the surface of the sphere which is in contact
with the bubble." My eye was drawn to the word "surface," hoping that
you meant surface and not volume.  Of course, either could be
calculated.

So that I can make my terminology clear, imagine a circle of radius a
(a cross-section of the container). Draw a horizontal line ABOVE the
center of this circle. Suppose the horizontal line is h units from the
top. I stress above since formulas may change if the horizontal line
is below the center. I assumed above since the bubble is probably
relatively small. Let me know if this isn't the case.

The chunk of the container lying above the plane corresponding to the
line is called a spherical segment. Its volume is V=(pi*h^2/3)(3a-h)
and its (top) surface area is S = 2*pi*a*h.

You said that the bubble doesn't change volume as it deforms. Okay, so
whatever its shape, its volume is (4/3)*pi*b^3. So,

(4/3)*pi*b^3 = (pi*h^2/3)(3a-h),

where a and b are known. We can solve for h (it's just a quadratic)
and, then, knowing h we can calculate S = 2*pi*a*h.

I'd be grateful if you would send a follow-up message.
For one thing, I'd like to see if you agree with my calculations and
my assumption about the size of the bubble. I'd also appreciate more
I'm co-authoring a calculus book and would like to use this example as
a project in the book. So, if I can attach some genuine application
words and background, all the better.

-Doctor Jerry,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
College Higher-Dimensional Geometry

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