An ellipsoid has radii with lengths of #3 #, #6 #, and #5 #. A portion the size of a hemisphere with a radius of #2 # is removed form the ellipsoid. What is the volume of the remaining ellipsoid?

Question

An ellipsoid has radii with lengths of #3   #, #6  #, and #5  #. A portion the size of a hemisphere with a radius of #2  # is removed form the ellipsoid. What is the volume of the remaining ellipsoid?

Noah Atkinson · Accepted Answer

#340/3pi~~356" units"^3# #"the volume of the ellipsoid is"# #•color(white)(x)V_(color(red)"ellipsoid")=4/3pir_1r_2r_3# #V_("ellipsoid")=4/3pixx3xx6xx5=120pi# #"the volume of the hemisphere is"# #•color(white)(x)V_(color(red)"hemisphere")=2/3pir^3# #V_("hemisphere")=2/3pixx8=16/3pi# #V_("remaining")=120pi-16/3pi=340/3pi~~356" units"^3#

Claire Akin · Answer

undefined To find the volume of the remaining ellipsoid after removing a hemisphere with a radius of 2, first, calculate the volume of the original ellipsoid using the formula $ V = \frac{4}{3}\pi a b c $, where $ a $, $ b $, and $ c $ are the lengths of the semi-axes.

Next, calculate the volume of the removed hemisphere using the formula $ V = \frac{2}{3}\pi r^3 $, where $ r $ is the radius of the hemisphere.

Then, subtract the volume of the removed hemisphere from the volume of the original ellipsoid to find the volume of the remaining ellipsoid.