Answer to Question #280567 in Differential Equations for Muffin

Question

A certain artifact was found in 2020 to contain presently 83% of its original carbon-14. In what year did the artifact first exist? The half-life of carbon-14 is approximate 5,740 years.

Answer 1

The amount of time $t$ passed from the moment the artifact first existed can be found from the radioactive decay law:

$$N = N_0\cdot 2^{-\dfrac{t}{T}}$$

where $N/N_0 = 0.83$, $T = 5740years$. Expressing $t$, obtain:

$$t = -T\log_2\dfrac{N}{N_0} = -5740\cdot \log_20.83 \approx 1543years$$

Since now is year 2020, the artifact first existed in year $2020 - 1543 = 477$

Answer. 477.