Question

Situation:A researcher in North America discoversa fossile that contains 65% of its originalamount of C-14..-ktN=NoeNo inital amount of C-14 (at time=t = 0)N== amount of C-14 at time tk= 0.0001t = time, in yearsFind the age of the fossile to the nearest year.

Answer 1

We have been given the equation of the decay as

`\begin{gathered} N=N_0e^(-kt) \\ where\text{ } \\ N_0=initial\text{ amount of C-14 at time t} \\ N=amount\text{ of C-14 at time t = 65\% of N}_0=0.65N_0 \\ k=0.0001 \\ t=time\text{ in years = ?} \end{gathered}`

So we are looking for the time

Plugging the values into the equation, we have

`\begin{gathered} N=N_0e^(-kt) \\ 0.65N_0=N_0e^(-0.0001t) \\ e^(-0.0001t)=(0.65N_0)/(N_0) \\ e^(-0.0001t)=0.65 \end{gathered}`

Taking Ln of both sides, we have

`\begin{gathered} ln(e^(-0.000t))=ln(0.65) \\ -0.0001t=ln(0.65) \\ t=(ln(0.65))/(-0.0001) \\ t=4307.82916 \end{gathered}`

Hence the answer is 4308 to the nearest year