12.4.31 Various Olympic problems

Condition: A mirror manufacturer plans to launch a new series. The new mirrors will have an area of ​​1 m \( ^{2} \) and will be in the shape of a rectangle with a semicircle at the right and left ends (see picture). The mirror will be in a frame. The production of the curved part of the frame is 2 times more expensive per centimeter than the production of the straight part. What should the radius of the semicircle be so that the production of the frame is as cheap as possible? mirror