12.4.31 Various Olympiad Problems

Condition: The manufacturer of mirrors plans to launch a new series in production. New mirrors will have an area of 1 m \ ( ^{2} \) and will be in the form of a rectangle with a semicircle on the right and left end (see drawing). The mirror will be in the frame. The production of the curved part of the frame is 2 times more expensive by a centimeter than the production of a direct part. What should be the radius of the semicircle so that the production of the frame is the cheapest? mirror