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 $y = y 0 e ^ { - 0.0002 t }$. In how many years will there be $89 \%$ of the isotope left?
A) 550 years
B) 583 years
C) 1100 years
D) 244 years

Question

Quizplus · Accepted Answer

We want to solve for $t$ when $y = 0.89y_0$.
$$0.89y_0 = y_0 e^{-0.0002t}$$
Dividing both sides by $y_0$ gives
$$0.89 = e^{-0.0002t}$$
Taking the natural logarithm of both sides gives
$$\ln(0.89) = -0.0002t\ln(e) = -0.0002t$$
Solving for $t$ gives
$$t = \frac{\ln(0.89)}{-0.0002} \approx 582.67$$
Therefore, the answer is B) 583 years (rounded to the nearest year).

Solve the Problem 2%2 \%2% Per 100 Years ttt Represents Time in Years and Y Represents the Amount of Rate

Solve the Problem $2 \%$ Per 100 Years $t$ Represents Time in Years and Y Represents the Amount of Rate