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Question

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius of 3 is :

A
233
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B
23
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C
6
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D
3
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Solution

The correct option is B 23

Let AOB=θ and radius of sphere is r.
Radius of cylinder =rsinθ
and Height of cylinder =2rcosθ

Volume of cylinder, V=π(rsinθ)2×2rcosθ
V=π2r3sin2θcosθ
dVdθ=π2r3(2sinθcos2θsin3θ)

For maxima or minima, dVdθ=0
sinθ(2cos2θsin2θ)=0
tanθ=2
cosθ=13
Height of cylinder =2rcosθ=2×3×13=23

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