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?

Question

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?

Theodore Abad · Accepted Answer

#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 is essentially a sphere's volume that has been stretched or compressed in a few 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#

Aurora Ayala · Answer

undefined To find the remaining volume of the ellipsoid after removing a portion the size of a hemisphere with a radius of 8, you can calculate the volume of the ellipsoid and subtract the volume of the removed portion.

The formula for the volume of an ellipsoid is given by:

$$ V = \frac{4}{3} \pi abc $$

Where $a$, $b$, and $c$ are the lengths of the semi-axes of the ellipsoid.

Given that the radii lengths of the ellipsoid are 9, 11, and 4, respectively, we have $a = 9$, $b = 11$, and $c = 4$.

Now, we calculate the volume of the ellipsoid:

$$ V_{\text{ellipsoid}} = \frac{4}{3} \pi \times 9 \times 11 \times 4 $$

Next, we need to calculate the volume of the removed portion, which is a hemisphere with a radius of 8. The formula for the volume of a hemisphere is:

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

Where $ r = 8 $.

$$ V_{\text{hemisphere}} = \frac{2}{3} \pi \times 8^3 $$

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

$$ \text{Remaining volume} = V_{\text{ellipsoid}} - V_{\text{hemisphere}} $$

You can perform the calculations to find the numerical value of the remaining volume.