Find the ratio of the height of the cone to the diameter of the sphere.

Asked by pratikbharadia | 30th Dec, 2009, 11:40: AM

Expert Answer:

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

 

 

Answered by  | 31st Dec, 2009, 08:32: AM

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