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Question

Find the volume of the largest cone can be inscribed in a sphere of radius R.

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Solution


Let r be the base radius x is the distance O the center of the sphere from the base and V the volume of the are
Height h of the cone =R+x
V=13πr2h=π3(R2x2)(R+x)
=π3(R2+R2xRx2x2)
dVdx=π3[R22Rx3x2]
d2Vdx2=π3[2R6x]
For max or min VdVdx=0
R22Rx3x2=0
(R+x)(x3x)=0 2) x=R,x3 but xR
When x=R3d2Vdx2<0 V is max only when x=R3
Max V=13π(R2R29)(R+R3)=32πR381=827(43πR3)=827 (volume of sphere)

1221390_890074_ans_56404f6a1753413391f7bb876df8fc04.jpg

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