Question

If a radioactive isotope has half-life h, then the number of atoms in a sample is given by a(t)=a-2-/h where ao gives the number of atoms in the sample at time 0. The radio active decay of carbon-14 leads to clever way of determining the age of fossils and remnants of plants and animals. When an organism dies, the radioactive carbon- 14 in it decays, with half of it gone in 5,730 years. a. Suppose a mummified cat has a ratio of carbon-14 to carbon-12 that is 61% of the corresponding ratio for living organisms. Approximately how long ago did the cat die ?

Answer 1

The number of Carbon-14 atoms in the sample is given by $a(t)=a_0.2^{-t/h}$ . Carbon-14 is converted to Carbon -12 by decay.

It is given that

$\frac{a}{a_0-a}=0.61\Rightarrow \frac{a}{a_0}=\frac{0.61}{1.61}$

The corresponding time is

a(t) ao.2t/h a(t) 2t/h ao 1.61 2t/h 0.61 In (1.61 0.61. t = h- In 2 In 0.61 1.61 t = (5730)In 2 t = 8023

The cat died approximately ${\color{Blue} 8023 \textup{ years}}$ ago.