Question

This is the last question I have and need help w/:

Given exponential decay rate of 0.9%, how old is an animal bone that has lost 40% of its carbon-14?

Answer 1

Since you started with 100% and after certain time t you loss 40% which means you're left with 60%, so your initial value is 100% and your final value is 60%, your decay rate k=0.9%, convert all these percentages to decimal form:
\[N(t)=N_{0}e^{-kt}\rightarrow N(t)=.6, N_{0}=1, k=.009\]
\[.6=1*e^{-.009t}\rightarrow .6=e^{-.009t}\rightarrow \ln(.6)=\ln(e^{-.009t})\]
\[\ln(.6)=(-.009t)\ln(e)\rightarrow \ln(e)=1, so \rightarrow \ln(.6)=-.009t\]
\[\frac{\ln(.6)}{-.009}=t\]