The given quantity of metal is to be cost 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)dot`

Explanation

The volume of half-cylinder is,


`{V}=frac{1}{2} pi{ }^{2} {~h}`

`Rightarrow {h}=frac{2 {~V}}{pi {r}^{2}}`


total surface area `=` area of rectangular base `+` area of 2 semi circular ends `+` area of curved surface


`{S}={h} times 2 {r}+2 times frac{1}{2} pi {r}^{2}+frac{1}{2} times 2 pi {rh}`
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