find the altitude of a right circular cone of maximum curved surface area which can be inscribed in a sphere of radius R

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Let radius and height of the inscribed right circular cone be r and h respectively. In ABC, by Pythagoras theorem, h-R2+r2=R2h2+R2-2hR+r2=R2r2=2hR-h2Let curved surface area of cone be S.S=πrlS=πrh2+r2S2=π2r2h2+r2=π22hR-h2h2+2hR-h2S2=π22hR-h22hRS2=π24h2R2-2h3RZ = π24h2R2-2h3RDifferentiating both sides with respect to h, we get,dZdh= π28hR2-6h2RFor maxima or minima dZdh=0π28hR2-6h2R=08hR2-6h2R=0Dividing each term by hR we get,8R-6h=0h=43Rd2Zdh2 = π216hR - 6h2d2Zdh2h=4R3 π264R29 - 96R29 = -32πR29 <0hence, Z is maximum at h=43ROr S2 is maximum at h=43Ror S is maximum at h=43R.     Altitude of a right circular cone of maximum curved surface area which can be inscribed in a sphere of radius R is 43R.

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