Expert: Paul Klarreich - 12/10/2010

Question
I have a spherical oil tank. If the tank is a quarter full by depth (i.e. depth = 0.5r, what percentage of the tank is filled?  

Answer
Questioner:   Malcolm
Country:  United Kingdom
Category:  Calculus
Private:  No
 
Subject:  Volume of partially filled sphere
Question:  I have a spherical oil tank. If the tank is a quarter full by depth (i.e. depth = 0.5r, what percentage of the tank is filled?
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Assume the sphere is centered at O.

This can be considered a 'volume of revolution', which we will do by the 'disk' method.  If you have not reached that, stop reading.

Each disk has

radius = height = y.

thickness = dx

dVolume = pi y^2 dx

And x^2 + y^2 = r^2, so y^2 = r^2 - x^2

dV = pi(r^2 - x^2) dx

Now integrate from x = r/2 to r.

I think that should do it.