An ellipsoid has radii with lengths of #5 #, #12 #, and #7 #. A portion the size of a hemisphere with a radius of #6 # is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

Question

An ellipsoid has radii with lengths of #5   #, #12  #, and #7  #. A portion the size of a hemisphere with a radius of #6  # is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

Violet Aldrich · Accepted Answer

#=1307#

The Volume of an Ellipsoid with radii #=5,12 and 7#

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

#=pi/6(2times12)(2times5)(2times7)#

#=pi/6(24times10times14)#

#=560pi#

#=1759.3#

Volume of an Hemisphere#=2/3(pir^3)# where #r=6# is the radius #=2/3pi(6)^3#

#=2/3pitimes216#

#=144pi#

#=452.4#

Thus, the Ellipsoid's residual volume

#=1759.3-452.4#

#=1307#

Noah Atkinson · Answer

undefined To find the remaining volume of the ellipsoid after removing a portion the size of a hemisphere with a radius of 6, we first need to calculate the volume of the ellipsoid and then subtract the volume of the removed portion.

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

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

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

Given that the semi-major axes of the ellipsoid are 5, 12, and 7, respectively, we can substitute these values into the formula:

$$ V_{\text{ellipsoid}} = \frac{4}{3} \times \pi \times 5 \times 12 \times 7 $$

$$ V_{\text{ellipsoid}} = \frac{4}{3} \times \pi \times 420 $$

$$ V_{\text{ellipsoid}} = 560 \pi $$

Next, we need to calculate the volume of the removed portion, which is a hemisphere with a radius of 6. The formula for the volume of a hemisphere is given by:

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

Substituting the radius (6) into the formula:

$$ V_{\text{hemisphere}} = \frac{2}{3} \times \pi \times 6^3 $$

$$ V_{\text{hemisphere}} = \frac{2}{3} \times \pi \times 216 $$

$$ V_{\text{hemisphere}} = 144 \pi $$

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

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

$$ V_{\text{remaining}} = 560 \pi - 144 \pi $$

$$ V_{\text{remaining}} = 416 \pi $$

Therefore, the remaining volume of the ellipsoid after removing the portion the size of a hemisphere with a radius of 6 is $ 416 \pi $.