Question

A certain radioactive isotope has leaked into a small stream. Four

Answer 1

Ok so

for a thing that is P amount at first, with a half life of k days
after t days, the amount left is A

the equation is
$A=P( \frac{1}{2})^{ \frac{t}{k} }$

given
13% of P remained=A
t=400

replace A with 0.13P and t with 400


0.13P=$P( \frac{1}{2})^{ \frac{400}{k} }$
divide both sides by P
$0.13=( \frac{1}{2})^{ \frac{400}{k} }$
take the ln of both sides
$ln0.13=( \frac{400}{k})ln \frac{1}{2}}$
multiply both sides by k
$kln0.13=400ln \frac{1}{2}}$
divide oth sides by ln0.13
k=$\frac{400ln \frac{1}{2} }{ln0.13}$
evaluate
k=135.897 days


the half life is 135.897 days