Math, asked by divyanshsharma93, 5 months ago

Prove that the volume of the largest cone that can be inscribed is a sphere of radius R is 8/27.​

Answers

Answered by mdadnanshaikh952
0

Answer:

Let the centre of the sphere be O and radius be R. Let the height and radius of the variable cone inside the sphere be h and r respectively.

So, in the diagram, OA=OB=R,AD=h,BD=r

OD=AD−OA=h−R

Using Pythagoras Theorem in △OBD,

OB

2

=OD

2

+BD

2

⇒R

2

=(h−R)

2

+r

2

⇒R

2

=h

2

+R

2

−2hR+r

2

⇒r

2

=2hR−h

2

Volume of the cone V =

3

1

πr

2

h

=

3

1

π(2hR−h

2

)h

=

3

2πh

2

R

3

πh

3

For maximum volume,

dh

dV

=0

⇒0=

3

4πhR

−πh

2

⇒h=

3

4R

∴V =

3

2πR

9

16R

2

81

64πR

3

=

81

(96−64)πR

3

=

81

32πR

3

=

27

8

×

3

4

πR

3

We know that the volume of the sphere is V

s

=

3

4

πR

3

Therefore, V=

27

8

V

s

Similar questions