Radioactive element X decays to element Y with decay constant -0.3. Y decays to stable element Z with decay constant $- 0.2$ . What is the system of differential equations for the amounts, $x ( t ) , y ( t ) , z ( t )$ of the elements X, Y, Z, respectively, at time t.
A) $\frac { d x } { d t } = - 0.2 x , \frac { d y } { d t } = 0.2 x - 0.3 y , \frac { d z } { d t } = 0.2 y$
B) $\frac { d x } { d t } = - 0.2 x , \frac { d y } { d t } = 0.2 x - 0.3 y , \frac { d z } { d t } = 0.3 y$
C) $\frac { d x } { d t } = - 0.3 x , \frac { d y } { d t } = 0.3 x - 0.2 y , \frac { d z } { d t } = 0.2 y$
D) $\frac { d x } { d t } = - 0.3 x , \frac { d y } { d t } = 0.3 x - 0.2 y , \frac { d z } { d t } = 0.3 y$
E) $\frac { d x } { d t } = - 0.3 y , \frac { d y } { d t } = 0.3 x - 0.2 y , \frac { d z } { d t } = 0.2 y$

Question

Quizplus · Accepted Answer

The decay of X to Y is characterized by a decay constant of -0.3, so $\frac{dx}{dt} = -0.3x$. Y decays to Z with a decay constant of -0.2, so $\frac{dy}{dt} = 0.3x - 0.2y$ (the rate of increase of Y is proportional to the amount of X decaying, and the rate of decrease of Y is proportional to its own amount). Finally, Z increases as Y decays, so $\frac{dz}{dt} = 0.2y$.

Radioactive Element X Decays to Element Y with Decay Constant −0.2- 0.2−0.2

Radioactive Element X Decays to Element Y with Decay Constant $- 0.2$