Answer to Question #229233 in Differential Equations for samsam

Question

Initially 100milligrams of a radioactive substance was present. After 6 hours the mass has decreased by 3%. The rate of decay is proportional to the amount of the substance present at time t. Determine the half-life of the radioactive substance. 

Answer 1

Given:

$m_0=100\:\rm mg$

$m=97\:\rm mg$

$t=6\:\rm hr$

The radioactive decay equation (https://en.wikipedia.org/wiki/Radioactive_decay) says

$$N=N_0*2^{-t/t_{1/2}}$$

The mass of substance is proportional to the number of atoms

$$m=\frac{N}{N_A}M$$

So

$$m=m_0*2^{-t/t_{1/2}}$$

Hence, the half-life of the radioactive substance

$$t_{1/2}=-t/\log_2\frac{m}{m_0}\\=-6\:\rm hr/\log_2\frac{97}{100}=137\: hr=5.7 \: days$$