In 170 days the number of radioactive nuclei decreases to on

In 170 days the number of radioactive nuclei decreases to one-eighth the number present initially. What is the half-life (in days) of the material?

Solution

given ( A / A_o ) = 1/8

t = 170 days

using equation A = A_o e^kt  

( A / A_o ) = e^kt  

( 1/8 ) = e^kX170

ln (1/8 ) = 190 X k X ln e

- 2.07944 = 190 X k X 0.999

k = - 2.07944 / 190 X 0.999

k = - 0.010955

and here half time ( A / A_o ) = 1/2

( A / A_o ) = e^{-0.010955 t}

1/2 = e-^{0.010955 t}

ln0.5 = - 0.010955 X t X lne

-0.693 = - 0.010955 X t X 0.999

t = 0.693 / 0.010955 X 0.999

t = 63.32 days