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

Question

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

Caroline Aucoin · Accepted Answer

The remaining volume is #=1380.2# The volume of an ellipsoid is #V_e=(4/3)piabc# The volume of a hemisphere is #V_h=(2/3)pir^3# The remaining volume #V_r=V_e-V_h# #V_r=((2pi)/3)(2abc-r^3)# #V_r=((2pi)/3)(2*8*7*7-5^3)# #=(1318pi)/3=1380.2#

Autumn Armstrong · Answer

#(1318 pi)/3# Volume of an ellipsoid equals #4/3 pi abc#. In this case it would be #4/3 pi *8*7*7= (1568 pi)/3 # Volume of an hemisphere is #2/3 pi r^3 #. In this case it would be #2/3 pi 5^3= (250 pi)/3# The volume of the remaining solid would be #(1568pi)/3 - (250 pi)/3 =(1318 pi)/3#

Carson Arnold · 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 hemisphere.

The formula for the volume $ V $ of an ellipsoid with semi-axes $ a $, $ b $, and $ c $ is given by:

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

Given the semi-axes lengths of the ellipsoid as $ a = 8 $, $ b = 7 $, and $ c = 7 $, we have:

$$ V_{\text{ellipsoid}} = \frac{4}{3} \pi (8)(7)(7) $$

Now, let's calculate the volume of the removed hemisphere. The formula for the volume $ V_h $ of a hemisphere with radius $ r $ is:

$$ V_h = \frac{2}{3} \pi r^3 $$

Given $ r = 5 $, we have:

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

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

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

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

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

$$ V_{\text{remaining}} \approx 1054.73 $$

So, the remaining volume of the ellipsoid after removing the hemisphere is approximately $ 1054.73 $ cubic units.