Question 36023This question is from textbook geometry
: A sphere is inscribed in a cone with radius 6 and height 8. Find the volume of the sphere. This question is from textbook geometry
You can put this solution on YOUR website! A sphere is inscribed in a cone with radius 6 and height 8. Find the volume of the sphere.
DRAW A SECTION...TRIANGLE ABC REPRESENTS THE SECTION OF THE CONE,WITH A AS VERTEX AND BC AS BASE.HENCE BC =6+6=12
DRAW AD PERPENDICULAR FROM A TO BC.AD IS HEIGHT OF CONE =8
DRAW A CIRCLE WITH CENTRE AT O ON AD
AS SECTION OF SPHERE TOUCHING BC,CA,AB AT D,E,F.
HENCE OD=OE=OF=RADIUS OF SPHERE=R SAY
NOW TRIANGLES ADC,AND AEO ARE RIGHT ANGLED
ANGLE ADC=90=ANGLE AEO...AS AEC IS TANGENT..SPHERE TOUCHING CONE.
ANGLE DAC=ANGLE OAE=SAME ANGLE
HENCE THE 2 TRIANGLES ADC AND AEO ARE SIMILAR.
HENCE DC/EO=AD/AE
DC=6.....AD=8......EO=R=RADIUS OF SPHERE.....AE=SQRT(AO^2-OE^2)
=SQRT{(AD-OD)^2-R^2}
=SQRT{(8-R)^2-R^2}=SQRT{64+R^2-16R-R^2}
=SQRT(64-16R)
HENCE WE HAVE
6/R=8/SQRT(64-16R)
6SQRT(64-16R)=8R
6*4SQRT(4-R)=8R
3SQRT(4-R)=R....SQUARING....
9(4-R)=R^2
R^2+9R-36=0
(R-3)(R+12)=0
R=3
