Question

A process creates a radioactive substance at the rate of 2 g/hr and the substance decays at a rate proportional to its mass, with constant of proportionality k=0.1(hr)^-1. If Q(t) is the mass of the substance at time t, find the limit of Q(t) as t approaches infinity.

Answer 1

@SithsAndGiggles

Answer 2

The constant, is it \(k=0.1\dfrac{1}{\text{hr}}\), as in "inverse-hour"?

Answer 3

The material is created at a rate of \(2\dfrac{\text{g}}{\text{hr}}\), which means at any given time, the rate is \(\dfrac{dQ}{dt}=2\).

You're told that this material also decays at a rate proportional to the current amount \(Q(t)\), which means you have to add a term that reflects this to get the ODE
\[\frac{dQ}{dt}=2+kQ\]
In this case, \(k=-0.1\) because it's a decay factor, and so
\[\frac{dQ}{dt}=2-0.1Q\]

Answer 4

Thank you!