At the start of the episode, Kirk states that they will meet the Enterprise in 4 hours, 21 minutes. Assuming they've been in flight for a short while already, that would probably mean about a five-hour trip to meet the Enterprise. Coupled with a 5-hour trip to pick up the commissioner, that would be a 10-hour shuttle ride. That's an awfully long trip for a shuttle to make. Considering the Miss Hedford's sickly condition, it would be a lot quicker to use the Enterprise to pick her up. Arguably the Enterprise was busy doing something else, but then why does Scotty spend all the episode looking for Kirk? Also, in "The Menagerie", Kirk indicates that the shuttle does not have more than a few hours of oxygen in it, so it seems that a five-hour shuttle ride would be pushing it, and that's assuming that they were able to refuel before they headed back.
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