An ellipsoid has radii with lengths of #8 #, #9 #, and #2 #. A portion the size of a hemisphere with a radius of #5 # is removed form the ellipsoid. What is the volume of the remaining ellipsoid?

Question

An ellipsoid has radii with lengths of #8   #, #9  #, and #2  #. A portion the size of a hemisphere with a radius of #5  # is removed form the ellipsoid. What is the volume of the remaining ellipsoid?

Theodore Abad · Accepted Answer

Remaining volume of the ellipsoid is #(326pi)/3# Volume of an ellipsoid with dimensions #a#, #b# and #c# is given by #4/3pixxaxxbxxc#, hence volume of ellipsoid of radii with lengths of #8#, #9# and #2# is #4/3pixx8xx9xx2=192pi#. Now as volume of sphere is given by #4/3pir^3#, volume of hemisphere of radius #5# cut from this is #1/2xx4/3pixx5^3=2/3xx125pi=(250pi)/3# Hence, remaining volume of the ellipsoid is #192pi-(250pi)/3=(326pi)/3#

Caroline Aucoin · Answer

undefined To find the volume of the remaining ellipsoid after removing a hemisphere with a radius of 5, we first 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.

Given that the semi-axes of the original ellipsoid are 8, 9, and 2, we have:

$a = 8$, $b = 9$, $c = 2$.

Next, we need to calculate the volume of the hemisphere that was removed. The formula for the volume of a hemisphere is:

$$V_{\text{hemi}} = \frac{2}{3}\pi r^3$$

where $r$ is the radius of the hemisphere. Given that the radius of the hemisphere is 5, we have:

$r = 5$.

Now, we can calculate the volume of the hemisphere:

$$V_{\text{hemi}} = \frac{2}{3}\pi (5)^3$$

Finally, we subtract the volume of the hemisphere from the volume of the original ellipsoid to find the volume of the remaining ellipsoid:

$$V_{\text{remaining}} = V_{\text{ellipsoid}} - V_{\text{hemi}}$$