CBSE Class 12-science Answered
Dear Student
Take Radius of Sphere = r
Height of cone = h
And θ is as shown in the figure.
We rae going to find the cone volume in terms r and θ. Then equate its derivative with respect to θ to zero it gives tha θ value at which cone valume is maximum. One we have theta. we can find h/ D. (D=2r)
h= r+rcos(θ)=r(1+cosθ) (1)
Volume of cone V= 1/3 ( Base radius2 X h) = 1/3[ Π r2 sin2θ X r(1+cosθ) ]=1/3[Π r3 (sin2θ+sin2 θ cosθ)
Take the derivative of V with respect to θ
dV=1/3 Π r3 { 2sin θ cosθ +[ sin2θ(-sin θ) + cosθ ( 2sin θ cosθ ) ] }
Equate the above equation to zero
2sin θ cosθ - sin3θ +cos 2θ sin θ = 0 ==> 2 cosθ +2cos 2θ = sin2 θ
2 cosθ (1+cosθ ) = 1-cos 2θ
= (1+cosθ )(1-cosθ )
2 cosθ= (1-cosθ ) ==> 3cosθ=1 and cosθ=1/3
h/d= r(1+cosθ) / 2r =(1+cosθ)/2 =( 1 + 1/3 ) / 2 = 4/3 X 2 = 8/3
h/d = 8/3
Regards
Topper learning team