Solve the problem.
-A certain radioactive isotope decays at a rate of $1 \%$ per 400 years. If $t$ represents time in years and y represents the amount of the isotope left, use the condition that $\mathrm { y } = 0.99 \mathrm { y } 0$ to find the value of $\mathrm { k }$ in the equation $\mathrm { y } = \mathrm { y } _ { 0 } \mathrm { e } ^ { \mathrm { kt } }$.
A) 0.00003
B) -0.00003
C) -0.01005
D) 0.00673

Question

Quizplus · Accepted Answer

The decay of a radioactive isotope can be modeled by the equation $y = y_0 e^{kt}$, where $y$ is the amount of the isotope remaining, $y_0$ is the initial amount, $k$ is the decay constant, and $t$ is time. Given that the isotope decays by $1\%$ in 400 years, we have $y = 0.99y_0$ when $t = 400$. Substituting these values into the decay equation gives $0.99y_0 = y_0 e^{400k}$. Dividing both sides by $y_0$ and taking the natural logarithm of both sides gives $\ln(0.99) = 400k$. Solving for $k$ yields $k = \frac{\ln(0.99)}{400}$, which is approximately $-0.000025$, but the closest option provided is $-0.00003$.

Solve the Problem 1%1 \%1% Per 400 Years ttt Represents Time in Years and Y Represents the Amount of Rate

Solve the Problem $1 \%$ Per 400 Years $t$ Represents Time in Years and Y Represents the Amount of Rate