Radioactive Isotopes, exponential decay

Question

The radioactive isotope 14C has a half-life of approximately 5715 years. Now there are 30g of 14C.

(1) How much of it remains after 1400 years? ____ g
 
(2) How much of it remains after 14000 years? ____ g

Answer 1

From y = y₀e^kt, model the equation 0.5y₀ = y₀e^k(5715). Then obtain 5715k = ln 0.5 or k = ln 0.5/5715.

After 1400 years, the remaining amount is given by y₁₄₀₀ equals 30e^{(1400ln 0.5/5715)} or 25.31501119 grams, equivalent to 25.315 grams.

After 14000 years, the remaining amount is given by y₁₄₀₀₀ equals 30e^{(14000ln 0.5/5715)} or 5.491486803 grams, equivalent to 5.491 grams.