Question #75d5c

1 Answer
Feb 11, 2018

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