What can we learn from observing a Cepheid variable in a distant galaxy that brightens and dims with a period of about 10 days?

Observing a Cepheid variable in a distant galaxy allows us to determine the distance to that galaxy based on the relationship between the star's pulsation period and its intrinsic brightness. Cepheid variables have a well-defined period-luminosity relationship, meaning that the time it takes for one cycle of brightness to dim and brighten is directly correlated with the star's absolute luminosity.

When the period of a Cepheid, such as the one with a 10-day cycle, is measured, astronomers can use that information to calculate the star's intrinsic brightness. By comparing this intrinsic brightness with the apparent brightness (how bright it appears from Earth), we can apply the inverse square law of light to determine the distance to the galaxy hosting the Cepheid variable. This technique has been fundamental in establishing the scale of the universe and measuring distances to galaxies far beyond our own.

The other options do not directly relate to what can be learned from observing the Cepheid's brightness changes. While the age and size of the variable star or galaxy might be inferred indirectly in broader studies of stellar populations, they are not the primary data obtained from observing a Cepheid. Similarly, assessing the chemical composition or pinpointing a galactic location are not tasks that can be solved just