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Magic Island

Magic Island is an old time radio show aimed at a juvenile audience. It mixed pulp adventure and science fiction elements. The series was syndicated in 1935 and had 130 15-minute episodes. Episodes are hosted on the Internet Archive.

From Wikipedia:

The storyline followed wealthy Patricia Gregory as she ended her 14-year search in the South Pacific for her long lost daughter Joan. When Joan was one-year-old, the Gregory yacht was shipwrecked near the 30th parallel south. Lashed to a lifeboat, Joan was presumed lost by all but Patricia. In the opening episode, she receives a tip about a white girl living on a mysterious island populated by scientifically advanced people in the Pacific. After the reunion of mother and daughter, the program dramatized their subsequent adventures on the island which could submerge to escape detection.