What is the half life of carbon 14 to the nearest 100 years?
About 5700 years.
With that short a half-life compared to Earth's age, carbon-14 "should be" long gone. But it keeps getting regenerated when cosmic rays hit nitrogen high in the atmosphere. We call this process cosmogenesis.
The cosmogenic carbon-14 circulates down to the Earth's surface where organisms take it in, keeping up a concentration of C-14 throughout their lifetimes. When, however, the organism dies or part of it is cut off, that material no longer replenishes its C-14 and the amount of that isotope drops with decay. By measurement of how much has decayed with the 5700-year half-life, carbon-14 can be used to date organic material.
See more about carbon-14 at https://en.wikipedia.org/wiki/Carbon-14.