In a nuclear disaster, there are multiple dangerous radioactive isotopes that can be detected. If $93.6 \%$ of a particular isotope emitted during a disaster was still present 6 years after the disaster, find the continuous compound rate of decay of this isotope
The continuous compound rate of decay of this isotope is (Round to six decimal places as needed.)
Answer
Final Answer: The continuous compound rate of decay of this isotope is approximately \(\boxed{-1.10233\%}\) per year.
Step 1 ：In a nuclear disaster, there are multiple dangerous radioactive isotopes that can be detected. If 93.6% of a particular isotope emitted during a disaster was still present 6 years after the disaster, we are asked to find the continuous compound rate of decay of this isotope.
Step 2 ：The continuous compound rate of decay of this isotope can be calculated using the formula for continuous compound interest, which is \(A = P * e^{rt}\), where \(A\) is the final amount, \(P\) is the initial amount, \(r\) is the rate of decay, and \(t\) is the time.
Step 3 ：In this case, we know that \(A\) is 93.6% of \(P\) (since 93.6% of the isotope is still present), and \(t\) is 6 years. We can rearrange the formula to solve for \(r\): \(r = \ln(A/P) / t\).
Step 4 ：Substituting the given values into the formula, we get \(A = 93.6\), \(P = 100\), and \(t = 6\).
Step 5 ：Solving for \(r\), we get \(r = -1.10233\).
Step 6 ：The rate of decay calculated is negative, which makes sense because the isotope is decaying over time, not growing. The rate of decay is approximately -1.10233% per year.
Step 7 ：Final Answer: The continuous compound rate of decay of this isotope is approximately \(\boxed{-1.10233\%}\) per year.
