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# A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends.Show that in order that the total surface area may be minimum, the ratio of the length of the cylinder to the diameter of its semi-circular ends is pi:(pi+2)

Arun Kumar IIT Delhi
7 years ago
Hi

$\\V =\pi r^2 h/2 \\h = 2V/(\pi r^2) \\S = \pi r^2+\pi rh+2rh \\S = \pi r^2 + 2V/r + 4V/(\pi r) = \pi r^2 + 2 (\pi+2) V / (\pi r) \\dS/dr = 2 \pi r - 2 (\pi + 2) V / (\pi r^2) = 0 \\\pi^2 r^3 = (\pi + 2) V \\=>h/2r=V/ (\pi r^3)=\pi / (\pi + 2)$

Thanks & Regards, Arun Kumar, Btech,IIT Delhi, Askiitians Faculty