Problema Solution
A certain radioactive isotope has leaked into a small stream. One hundred days after the leak, 6% of the original amount of the substance remained. Determine the half-line of this radioactive isotope.
Answer provided by our tutors
The amount x after t time is given by:
x(t)=x0*e^(−kt)
for some constant k.
We know that
x(100)=0.06*x0 or
x0*e^(−100k)=0.06x0
e^(−100k)=0.06 use the natural log and solve for k
k = (-1/100) ln(0.06)
k = 0.02813
We need to find the half-life, that is, t such that
x0*e^(−kt)=0.5*x0
e^(−0.02813t)=0.5
t = ln(0.5)/(−0.02813)
t = 24.64 years
The half-life is 24.64 years.