maths

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 $=R+x$

$∴$ $V=31 πr_{2}h=3π (R_{2}−x_{2})(R+x)$

$=3π (R_{2}+R_{2}x−Rx_{2}−x_{2})$

$∴$ $dxdV =3π [R_{2}−2Rx−3x_{2}]$

$dx_{2}d_{2}V =3π [−2R−6x]$

For max or min $VdxdV =0$

$∴$ $R_{2}−2Rx−3x_{2}=0$

$⇒(R+x)(x−3x)=0$ 2) $x=−R,3x $ but $x=−R$

When $x=3R dx_{2}d_{2}V <0$ $V$ is max only when $x=3R $

$∴$ Max $V=31 π(R_{2}−9R_{2} )(R+3R )=8132πR_{3} =278 (34 πR_{3})=278 $ (volume of sphere)

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