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a given quantity of metal is to be cast in to a solid half circular cylinder (with 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 circular ends is pi:pi+2?
If r is the radius and h the height then the total volume (which is fixed) is V = ? r2 h/2 so
h = 2V/(? r2)   -----------------(1)

 The surface area is S = ?r2+?rh+2rh so, replacing h

S = ? r2 + 2V/r + 4V/(?r) = ? r2 + 2 (?+2) V / (? r) and so
dS/dr = 2 ? r - 2 (? + 2) V / (? r^2) = 0 so
?2 r3 = (? + 2) V  -------------(2)

The ratio of the length of the cylinder to the diameter of its circular ends is h/(2r) which, from (1) is equal to V/ (? r3) and, from (2) ? / (? + 2)


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