40. 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 w `(

Options

A. `pi:(pi+2)`
B. `(pi+2) : pi`
C. `1:1`
D. none of the above

Answer

A

Explanation

volume of half cylinder with height '1' and radius 'r' `=1/2 pi r^(2)l`
surface area of half cylinder

`S=lxx2r+2(1/2 pir^(2))+2(1/2pirl)`
`s=(4v)/(pir)+(pi.2v)/(pi r)+pir^(2)`
`(ds)/(sdr) rarr (-4v)/(pi r^(2))-(2v)/(r^(2))+2pi r =0` ltrbgt `rarr 2pi r = (4v)/(pi r^(2)) +(2v)/(r^(2))`
`rarr 2pi r = (4.pir^(2)l)/(2pi r&^(2))+(2)/(r^(2)).1/2pir^(2)l`
`rarr (l)/(2r)=(pi)/(pi+2)`