Final answer:
A Cepheid star with a 10-day brightness variation cycle has a luminosity several thousand to 10,000 times that of the Sun, according to the period-luminosity relationship.
Step-by-step explanation:
The approximate luminosity of a Cepheid star that varies in brightness on a 10-day cycle can be learned through the period-luminosity relation. This important astronomical relationship indicates that the longer the period of luminosity variation, the greater the average luminosity of the Cepheid variable. As most cepheids have periods ranging from 3 to 50 days and luminosities about 1000 to 10,000 times greater than that of the Sun, a Cepheid with a 10-day period would have a luminosity on the higher end of this range. Therefore, we can conclude that a Cepheid star with a 10-day period has an approximate luminosity ranging from several thousand to possibly 10,000 times that of the Sun's luminosity.