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Prove that the radius of the right circular cylinderof greatest curved surface area which can beinscribed in a given cone is half of that of thecone. |
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Answer» H = height of the cone , R = radius of the coneh = height of the cylinder, r = radius of the cylinder,S = lateral surface area of the cylinder S = 2π r hr = R (1 - h/H) . S = 2π (R (1 - h/H)) hdS/dh = 2π (R/H) (H - 2h)d²S/dh² = -4π (R/H)H - 2h = 0 .......... set dS/dh = 0 to find the stationary pointsh = H/2 .............. as d²S/dh² < 0 r = R (1 - (H/2)/H) ....... plug h=H/2 into r = R (1 - h/H)r = R/2 |
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