A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order for the total surface area to be minimum, the ratio of the length of the cylinder to the diameter of its semi-circular ends is `pi:(pi+2)`.

Explanation

To solve the problem, we need to find the ratio of the length of a half cylinder to the diameter of its semicircular ends such that the total surface area is minimized. Let's denote:

- \( r \) = radius of the semicircular ends
- \( l \) = length of the cylinder
- \( d \) = diameter of the semicircular ends = \( 2r \)

### Step 1: Volume of the Half Cylinder
The volume \( V \) of the half cylinder can be expressed as:
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