Question

An ellipsoid has radii with lengths of `5`, `5`, and `8`. A portion the size of a hemisphere with a radius of `3` is removed form the ellipsoid. What is the remaining volume of the ellipsoid?

Answer 1

see a solution step below;

Explanation:

Please note that we are dealing with an Ellipsoid and Hemisphere..

The formula for the volume of an ellipsoid is given below;

`"Volume of an ellipsoid" rArr V_e = 4/3piabc`

The formula for the volume of a hemisphere is given below;

`"Volume of a hemisphere" rArr V_h = 2/3pir^3`

Since we are removing a portion of a Hemisphere from an Ellipsoid, it means that we need to subtract the volume of a hemisphere from the ellipsoid which is given below;

`"Remaining volume of an ellipsoid" color(white)x R_(ve) = V_e - V_h`

`R_(ve) = V_e - V_h`

Where;

`R_(ve) = "Remaining volume of an ellipsoid"`

`V_e = "Volume of an ellipsoid"`

`V_h = "Volume of a hemisphere"`

`a, b, c = "lengths"`

`r = "radius"`

`R_(ve) = V_e - V_h`

`V_e = 4/3piabc`

`V_h = 2/3pir^3`

`a, b, c = 5, 5, 8 "respectively"`

`r = 3`

`pi = 3.142`

Hence substituting the values into the formula;

`R_(ve) = V_e - V_h`

`R_(ve) = 4/3pi(5 xx 5 xx 8)- 2/3pi(3)^3`

`R_(ve) = 4/3pi(200)- 2/3pi(27)`

`R_(ve) = (4(200))/3 pi - (2(27))/3 pi`

`R_(ve) = 800/3 pi - 54/3 pi`

`R_(ve) = (800 - 54)/3 pi`

`R_(ve) = 746/3 pi`

`R_(ve) = 248.67 pi`

`R_(ve) = 248.67 xx 3.142`

`R_(ve) = 781.31cm^3`

Therefore the remaining volume of the ellipsoid is `781.31cm^3`