Solve the problem.
-If the half-life of an element is $67 \mathrm { yr }$ and the initial quantity is $3 \mathrm {~kg}$, write a function of the form $Q ( t ) = Q _ { 0 } e ^ { - k t }$ to model the quantity of the element left after $t$ years. Round $k$ to 4 decimal places.
A) $Q ( t ) = 67 e ^ { - 0.0103 t }$
B) $Q ( t ) = 67 e ^ { - 0.0034 t }$
C) $Q ( t ) = 3 e ^ { - 0.0034 t }$
D) $Q ( t ) = 3 e ^ { - 0.0103 t }$

Question

Quizplus · Accepted Answer

The half-life formula is $Q(t) = Q_0 e^{-kt}$, where $Q_0$ is the initial quantity, $k$ is the decay constant, and $t$ is time. For a half-life of 67 years, $Q(t) = \frac{Q_0}{2}$ when $t = 67$. Substituting these values into the formula gives $0.5 = e^{-67k}$. Solving for $k$ gives $k \approx 0.0103$. Therefore, the correct model with $Q_0 = 3 \, \text{kg}$ is $Q(t) = 3e^{-0.0103t}$.

Solve the Problem 67yr67 \mathrm { yr }67yr And the Initial Quantity Is

Solve the Problem $67 \mathrm { yr }$ And the Initial Quantity Is