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S is the total surface area of the closed right circular cylinder whose radius is r, height is h, and volume is V.Then surface area S = 2 * pi * r* h + pi * r^2 * h ------ (i)V = ½ [Sr – 2*pi*r^3]Therefore, dV / dr = ½ [S – 6 * pi * r^2 ]For extrema (maxima or minima) dV/dr=0Therefore S – 6 * pi * r^2 = 0 S – 6 * pi * r^2 = 0Plugging this value of S in eq.(i),we get
h (at V-extrema)= 2rFor maxima, d2V / dr2 = – 6* pi*r = negative Therefore, V is maximum when height, h = 2r = diameter of the base.
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