Answer to Question #228817 in Calculus for frost

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

The rate of decay is proportional to the amount of the substance present at time t

This means that the amount of the substance can be modeled by the following differential equation:

$\frac{dQ}{dt}=-rt$

Which has the following solution:

$Q{(t)}=Q(0)e^{-rt}$

In which Q(t) is the amount after t hours, Q(0) is the initial amount and r is the decay rate.

After 6 hours the mass had decreased by 3%.

This means that . We use this to find r.

This means that $Q(6)=(1-0.03)Q(0)=0.97Q(0)$

We use this to find r

$Q{(t)}=Q(0)e^{-rt}$

$0.97Q(0)=Q(0)e^{-6r}$

$e^{-6r}=0.97$

$ln(e^{-6r})=ln(0.97)$

$-6r=ln0.97$

$r=-\frac{ln0.97}{6}$

$r=0.0051$

$Q{(t)}=Q(0)e^{-0.0051t}$

Determine the half life of radio active substance.

This is t for which $Q(t)=0.5Q(0)$. So

$Q(t)=Q(0)e^{-0.0051t}$

$0.5Q(0)=Q(0)e^{-0.0051t}$

$e^{-0.0051t}=0.5$

$ln(e^{-0.0051t})=ln(0.5)$

$-0.0051t=ln0.5$

${t}=\frac{0.5}{0.0051}$

$t=135.9$

The half-life of the radioactive substance is 135.9 hours