How do we know the half-life of carbon-14 is 5730 years?

Radiocarbon dating is the process of determining the age of a sample by examining the amount of 14C remaining against its known half-life, 5,730 years. However, when an organism ceases to exist, it no longer takes in carbon from its environment and the unstable 14C isotope begins to decay.

What percentage of carbon-14 remains after 3 half-lives?

After 3 half-life, 12 of the 14 of the C-14 = 18 of the C-14 would remain. After n half-lives, 12n of the C-14 would remain.

How much would 100g of carbon-14 be after 11460 years?

The currently accepted value for the half-life of 14C is 5,730 years. This means that after 5,730 years, only half of the initial 14C will remain; a quarter will remain after 11,460 years; an eighth after 17,190 years; and so on.