We ... on the beach for several hours by the time she reads our letter.

The correct answer is: will have been sunbathing Because: The sentence refers to an action that will be in progress for a period of time before another action happens in the future. Therefore, the future perfect continuous tense is used to express this idea. In this case, 'will have been sunbathing' indicates that the act of sunbathing will have been ongoing for several hours before the letter is read.