Question

A cone is circumscribed about a sphere of radius r. Show that the volume of the cone is maximum when its semi vertical angle is sin−1(13)

Answer 1

Answer:

Step-by-step explanation:

Given 'a' is the sphere radius of the sphere .

Given 'a' is the sphere radius of the sphere .

Let OD=x

ΔODC

∴DC2=a2−x2

Volume of the cone =13x(x2)(a+x)

=13π(a2−x2)(a+x)

differentiating with respect to x we get,

V′=13π[(a2−x2)(1)+(a+x).(−2x)]

=13π[a2−x2−2ax−2x2]

=13π[a2−2ax−3x2]

V′′=13π[−2a−6x]<0

hence maximum

if V′=0

13π[a2−2ax−3x2]=0

=>a2−2ax−3x2=0

On factorizing,

(a−3x)(a+x)=0

=> a=3x

or xa=13=sinθ

θ=sin−1(13)

Height of the cone =a+x

=a+a3

=4a3

When the attitude is 4a3 and semi vertical angle is sin−1 (1/3)

Therefore. volume of the cone is minimum.