A solid hemisphsere is mounted on a solid cylinder, both having equla radii. If the whole solid is to have a fixed surface are and the maximum possible volume, then the ratio of the height of the cylinder to the common radius is

Options

A. `1:1`
B. `1:2`
C. `2:1`
D. `sqrt(2):1`

Answer

A

Explanation

`S=2 pi R^(2)+2 pi Rh +pi R^(2)`
(R= radius of hemisphere & cylinder)
`V=(2)/(3)pi R^(3)+piR^(2)H`
`V=(2)/(3)pr R^(3)+ piR^(2)xx((5-3 pi R^(2))/(2 pi R))`
`(dV)/(dR)= 2 pi R^(2)+(5)/(2)-(9 pi)/(2) R^(2)`
For maximum & minimum `(dV)/(dR)=0`
`5pi R^(2)=S`
`5piR^(2)=3piR^(2)+2 piRh`
`h:H=1:1`