An ellipsoid has radii with lengths of #9 #, #8 #, and #6 #. A portion the size of a hemisphere with a radius of #3 # is removed form the ellipsoid. What is the volume of the remaining ellipsoid?

Question

An ellipsoid has radii with lengths of #9   #, #8  #, and #6  #. A portion the size of a hemisphere with a radius of #3  # is removed form the ellipsoid. What is the volume of the remaining ellipsoid?

Aurora Ayala · Accepted Answer

The remaining volume is #=1753.1u^3#

The volume of an ellipsoid is #V_e=4/3abc# where #a#, #b#, and #c# are the lengths.

Here,

#a=9u#

#b=8u#

and #c=6u#

so,

#V_e=4/3pixx9xx8xx6=576pi=1809.6u^3#

The volume of a hemisphere is #V_h=2/3pir^3# where #r# is rhe radius

Here,

#r=3u#

so,

#V_h=2/3pir^3=2/3pixx3^3=56.5u^3#

The volume that is still there is

#V_r=V_e-V_h=1809.6-56.5=1753.1u^3#

Aurora Ayala · Answer

undefined To find the volume of the remaining ellipsoid after removing a hemisphere:

1. Calculate the volume of the original ellipsoid using the formula: $V = \frac{4}{3}\pi abc$, where $a$, $b$, and $c$ are the semi-axes of the ellipsoid.

2. Calculate the volume of the removed hemisphere using the formula: $V_{hemisphere} = \frac{2}{3}\pi r^3$, where $r$ is the radius of the hemisphere.

3. Subtract the volume of the hemisphere from the volume of the original ellipsoid to find the volume of the remaining portion.

Using the given semi-axes lengths: $a = 9$, $b = 8$, and $c = 6$, and the radius of the removed hemisphere: $r = 3$, you can calculate the volumes as described above.

Subtract the volume of the removed hemisphere from the volume of the original ellipsoid to obtain the volume of the remaining ellipsoid.