An ellipsoid has radii with lengths of #9 #, #11 #, and #4 #. A portion the size of a hemisphere with a radius of #8 # is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

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Answer 1 · 2016-03-12T19:42:19

#560/3 pi~~586.43#

The volume of an ellipsoid with radii #r_1#, #r_2# and #r_3# is:

#4/3 pi r_1 r_2 r_3#

since it's basically the volume of a sphere stretched/squashed in a couple of dimensions.

The volume of a hemisphere of radius #r# is:

#1/2*4/3 pi r^3 = 2/3 pi r^3#

In our example, #r_1 = 9#, #r_2 = 11#, #r_3 = 4# and #r = 8# so the volume remaining is:

#4/3 pi r_1 r_2 r_3 - 2/3 pi r^3#

#=2/3 pi (2*9*11*4 - 8^3)#

#=2/3 pi (792 - 512)#

#=2/3 pi * 280#

#=560/3 pi#

#~~586.43#