Solve the problem.
-A certain radioactive isotope decays at a rate of 2% per 100 years. If t represents time in years and y represents the amount of the isotope left then the equation for the situation is y = y0e^-0.0002t. In how many years will
There be 89% of the isotope left?

Question

Quizplus · Accepted Answer

The equation given is $y = y_0e^{-0.0002t}$, where $y_0$ is the initial amount of the isotope, $y$ is the amount left after $t$ years, and $0.0002$ is the decay rate per year. To find when 89% of the isotope is left, we set $y/y_0 = 0.89$ and solve for $t$:$$0.89 = e^{-0.0002t}$$Taking the natural logarithm of both sides:$$\ln(0.89) = -0.0002t$$Solving for $t$:$$t = \frac{\ln(0.89)}{-0.0002} \approx 583 \text{ years}$$Therefore, it will take approximately 583 years for 89% of the isotope to be left.

Solve the Problem. -A Certain Radioactive Isotope Decays at a Rate of 2