An ellipsoid has radii with lengths of #5 #, #12 #, and #8 #. A portion the size of a hemisphere with a radius of #7 # is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

Question

An ellipsoid has radii with lengths of #5   #, #12  #, and #8  #. A portion the size of a hemisphere with a radius of #7  # is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

Sadie Adams · Accepted Answer

#=1292.24#

The Volume of an Ellipsoid with radii #=5,12 and 8#

#=pi/6times#(major-axis)#times#(minor axis)#times#(vertcal-axis)

#=pi/6(2times12)(2times5)(2times8)#

#=pi/6(24times10times16)#

#=640pi#

#=2010.62#

Volume of an Hemisphere#=2/3(pir^3)# where #r=7# is the radius #=2/3pi(7)^3#

#=2/3pitimes343#

#=228.67pi#

#=718.4#

Thus, the Ellipsoid's residual volume

#=2010.62-718.4#

#=1292.24#

Noah Atkinson · Answer

undefined To find the remaining volume of the ellipsoid after removing a hemisphere, you can use the formula for the volume of an ellipsoid and subtract the volume of the removed hemisphere.

The formula for the volume of an ellipsoid is:

$$ V = \frac{4}{3} \pi a b c $$

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

Given that the radii lengths of the ellipsoid are $a = 5$, $b = 12$, and $c = 8$, the volume of the original ellipsoid is:

$$ V_{\text{original}} = \frac{4}{3} \pi (5)(12)(8) $$

To find the volume of the hemisphere that's removed, we use the formula for the volume of a hemisphere:

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

Where $r$ is the radius of the hemisphere, given as $r = 7$.

Substituting the values, we get:

$$ V_{\text{hemisphere}} = \frac{2}{3} \pi (7)^3 $$

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

$$ V_{\text{remaining}} = V_{\text{original}} - V_{\text{hemisphere}} $$

$$ V_{\text{remaining}} = \frac{4}{3} \pi (5)(12)(8) - \frac{2}{3} \pi (7)^3 $$

Calculate the values and you'll get the remaining volume of the ellipsoid.