Q. 254.3( 3 Votes )

Show that the vol

Answer :

Given: Height of cone is ‘h’ and semi-vertical angle of the cone is ‘a’


To prove: the volume of the greatest cylinder that can be inscribed in a cone is


Let PQR is cone and PO = h


Let x be the radius of cylinder ABCD inscribed in a cone PQR



Height of cylinder = OO’ = PO – PO’


In Δ APO’





PO’ = x cot a


Volume of cylinder, V


= πr2h


= πx2(h – x cot a)


= πx2h – πx3 cot a


We need to maximize volume


V = πx2h – πx3 cot a


Differentiating both sides with respect to x:





Now, Put :


2πxh – 3πx2 cot a = 0


πx(2h – 3xcot a) = 0


2h – 3xcot a = 0


– 3xcot a = -2h




Differentiating again both sides with respect to x:





From (1) :





Since,



V is maximum at


V = πx2h – πx3 cot a


Put :









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