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Question

The volume of the greatest cylinder which can be inscribed in a cone of height 30 cm and semi-vertical angle 30 is

A
4000π3 cubic cm
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B
400π3 cubic cm
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C
4000π3 cubic cm
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D
None of these
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Solution

The correct option is A 4000π3 cubic cm
From figure, we have
tan30=r30h
h=303r
The volume of cylinder, V=πr2h=πr2(303r)
Now, dVdr=0
π(60r33r2)=0
r=203
d2Vdr2=π(6063r)(d2Vdr2)r=203=60π<0
Hence, Vmax=π(203)2(303×203)=π4003×10
Vmax=4000π3

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