Question

Question #75d5c

Answer 1

`V2=3/4V1` has been proved as shown below

Explanation:

A cone circumscribes a sphere and has its slant height equal to the diameter of its base. Show that the volume of the cone is 3/4 the volume of the sphere?

Given:
Slant height equal to diameter of the base.
ie,
the cone has 2 slant heights and one base which are all equal, and forming an equilateral angle
The angle at the vertex is 60 degrees
Semi vertex angle is `1/2(60)=30 degrees`
If the slant height is `l`
radius of the circumscribed sphere is `r`
Angle between the radii is `theta=120 degrees`
Thus `l^2=r^2+r^2-2rrcos(120)`
`cos120=-1/2`
Now, `l^2=r^2+r^2-2rr(-1/2)`
`l^2=3r^2`
`l=(sqrt3)r`
Volume of a sphere is V1

`V1=4/3pir^3`
radius of the cone is `a=l/2`
`a^2=(l/2)^2`
`a^2=l^2/4`
`l^2=3r^2`
`a^2=3r^2/4`
Height of the cone is given by
`l^2=(l/2)^2+h^2`
`h^2=3l^2/4`
`h=(sqrt3)/2l`
`h=(sqrt3)/2sqrt3r`
`h=3/2r`
Volume of the cone is V2
`V2=1/3pia^2h`
`V2=1/3pi(3r^2)/4(3/2r)`
`V2=pir^3`

Thus,
Volume of sphere is `V1=4/3pir^3`
Volume of the cone is `V2=pir^3`
`V2=3/4(4/3)pir^3`
`V2=3/4V1`