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Question

Find the height of the right circular cylinder of maximum volume V which can be inscribed in a sphere of radius R.

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Solution

Radius of Cylinder =Rcosθ
Height of Cylinder =2Rsinθ
Volume of Cylinder =π(Rcosθ)2(2Rsinθ)



V=2πR3sinθcos2θ
To maximize volume
dVdθ=0
2πR3(cos2θ.cosθ2sin2θcosθ)=0
cos2θ.cosθ2sin2θcosθ)=0
cosθ(cos2θ2sin2θ)=0
cosθ[cos2θ2(1cos2θ)=0
cosθ(3cos2θ2)=0
cosθ=0θ=π2 (Not possible)
so, we must have 3cos2θ2=0
cosθ=23 i.e


Hence sinθ=13
Height of cylinder =2Rsinθ=2R3

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