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
4000π√3 cubic cm
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D
None of these
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Solution
The correct option is A4000π3 cubic cm From figure, we have tan30∘=r30−h ⇒h=30−√3r
The volume of cylinder, V=πr2h=πr2(30−√3r)
Now, dVdr=0 ⇒π(60r−3√3r2)=0 ⇒r=20√3 d2Vdr2=π(60−6√3r)⇒(d2Vdr2)r=20√3=−60π<0
Hence, Vmax=π(20√3)2(30−√3×20√3)=π4003×10 ⇒Vmax=4000π3