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Answer:
The half-life of the radioactive isotope is 346 years.
Explanation:
The decay rate of the isotope is modelled after the following first-order linear ordinary differential equation:
Where:
- Current isotope mass, measured in kilograms.
- Time, measured in years.
- Time constant, measured in years.
The solution of this differential equation is:
Where is the initial mass of the isotope. It is known that radioactive isotope decays at a yearly rate of 0.2 % annually, then, the following relationship is obtained:
The time constant associated to the decay is:
Finally, the half-life of the isotope as a function of time constant is given by the expression described below:
If , the half-life of the isotope is:
The half-life of the radioactive isotope is 346 years.