The amount of carbon-14 present in animal bones t years after the animal's death is given by $P(t)=P_{0} e^{-0.00012097 t}$. How old is an ivory tusk that has lost $36 \%$ of its carbon- 14 ? The ivory tusk is years old.
Solution
Step 1 ：The amount of carbon-14 present in animal bones t years after the animal's death is given by \(P(t)=P_{0} e^{-0.00012097 t}\). We are asked to find how old is an ivory tusk that has lost 36% of its carbon-14.
Step 2 ：This means that the amount of carbon-14 has decreased to 64% (100% - 36%) of its original amount. So, we need to solve the equation \(0.64P_{0} = P_{0} e^{-0.00012097 t}\) for t.
Step 3 ：We can simplify this equation by dividing both sides by \(P_{0}\), which gives us \(0.64 = e^{-0.00012097 t}\).
Step 4 ：To solve for t, we can take the natural logarithm of both sides. This gives us \(\ln(0.64) = -0.00012097 t\).
Step 5 ：Solving for t, we get \(t = \frac{\ln(0.64)}{-0.00012097}\).
Step 6 ：Calculating the above expression, we find that \(t \approx 3689.237849288414\).
Step 7 ：So, the ivory tusk is approximately \(\boxed{3689}\) years old.
