The correct domain of the function d(t) representing the distance traveled by the balloon based on time in hours is option (c) (0,478], including all the moments up to and including 478 hours but not starting exactly at 0.
The domain of a function refers to the set of all possible input values (usually representing time, in this case) for which the function is defined. Given that Bertrand Piccard and Brian Jones traveled a total of 478 hours and the distance is modeled by the function d(t) = 97.8t, where 't' is the time in hours, the domain of this function is the set of all ‘t’ values between 0 and 478, including 0 and 478.
Therefore, the correct option representing the domain is (0,478] which signifies that the starting time is just after 0 hours (since one doesn't start counting at zero), but includes 478 hours. Options involving 1999 and 97.8 are not relevant to the domain of time. The domain doesn't include the initial moment exactly at 0 since they started after 0 hours but includes the accumulation of all the moments leading up to and including 478 hours.