Question #75d5c

1 Answer
Feb 11, 2018

V2=3/4V1V2=34V1 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 degrees12(60)=30degrees
If the slant height is ll
radius of the circumscribed sphere is rr
Angle between the radii is theta=120 degreesθ=120degrees
Thus l^2=r^2+r^2-2rrcos(120)l2=r2+r22rrcos(120)
cos120=-1/2cos120=12
Now, l^2=r^2+r^2-2rr(-1/2)l2=r2+r22rr(12)
l^2=3r^2l2=3r2
l=(sqrt3)rl=(3)r
Volume of a sphere is V1

V1=4/3pir^3V1=43πr3
radius of the cone is a=l/2a=l2
a^2=(l/2)^2a2=(l2)2
a^2=l^2/4a2=l24
l^2=3r^2l2=3r2
a^2=3r^2/4a2=3r24
Height of the cone is given by
l^2=(l/2)^2+h^2l2=(l2)2+h2
h^2=3l^2/4h2=3l24
h=(sqrt3)/2lh=32l
h=(sqrt3)/2sqrt3rh=323r
h=3/2rh=32r
Volume of the cone is V2
V2=1/3pia^2hV2=13πa2h
V2=1/3pi(3r^2)/4(3/2r)V2=13π3r24(32r)
V2=pir^3V2=πr3

Thus,
Volume of sphere is V1=4/3pir^3V1=43πr3
Volume of the cone is V2=pir^3V2=πr3
V2=3/4(4/3)pir^3V2=34(43)πr3
V2=3/4V1V2=34V1