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

To solve the problem of finding the ratio of the length of a half-cylinder to the diameter of its semi-circular ends such that the total surface area is minimized, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( r \) = radius of the semi-circular ends
- \( h \) = length of the cylinder
- The diameter of the semi-circular ends = \( 2r \)

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