$\"\"$

Let the total length is $\"\"$ .

Redraw the figure :

$\"\"$

Observe the above diagram :

Total length of the power line is $\"\"$ .

Here $\"\"$ and $\"\"$ are constants.

$\"\"$

$\"\"$ .

Derivative on each side with respect to $\"\"$ .

$\"\"$

Equate derivative to zero.

$\"\"$

$\"\"$ .

$\"\"$

The total length of power line from the power supply to the factories is minimum when $\"\"$ .