Final answer:
Astronomers determine the age of the solar system to be around 4.5 billion years by using radioactive dating of meteorites, which are among the oldest, unchanged objects in the solar system, representing the time since the first solids condensed in the solar nebula.
Step-by-step explanation:
Astronomers estimate the age of the solar system to be about 4.5 billion years old primarily through radioactive dating of meteorites and rocks from the Earth and Moon. By analyzing the isotopic composition of these materials and applying knowledge of radioactive half-lives, scientists determine how many half-lives have elapsed since these primordial substances first solidified. Meteorites, which have not been altered significantly since their formation, are particularly valuable in this regard. The radioactive ages of these meteorites are found to be around 4.5 billion years, making them the oldest known objects in the solar system. This age is consistent with the age of the planetary system as it represents the time since the first solids condensed from the solar nebula and began to coalesce into larger bodies.