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.cosθ−2sin2θcosθ)=0 cos2θ.cosθ−2sin2θcosθ)=0 cosθ(cos2θ−2sin2θ)=0 cosθ[cos2θ−2(1−cos2θ)=0 cosθ(3cos2θ−2)=0 ∵cosθ=0⇒θ=π2 (Not possible)
so, we must have 3cos2θ−2=0 cosθ=√23 i.e