A drug is administered intravenously at a constant rate of $r$ mg/hour and is excreted at a rate proportional to the quantity present, with constant of proportionality $k>0$.

(Set up and) Solve a differential equation for the quantity, $Q$, in milligrams, of the drug in the body at time $t$ hours. Assume there is no drug in the body initially. Your answer will contain $r$ and $k$.
$Q =$

Graph $Q$ against $t$. What is $Q_{\infty}$, the limiting long-run value of $Q$?
$Q_{\infty} =$

If $r$ is doubled (to $2r$), by what multiplicative factor is $Q_{\infty}$ increased?
$Q_{\infty}$ (for $2r$) = $Q_{\infty}$ (for $r$)

Similarly, if $r$ is doubled (to $2r$), by what multiplicative factor is the time it takes to reach half the limiting value, $\frac{1}{2}Q_{\infty}$, changed?
$t$ (to $\frac12 Q_{\infty}$), for $2r$) = $t$ (to $\frac12 Q_{\infty}$), for $r$)

If $k$ is doubled (that is, we use $2k$ instead of $k$), by what multiplicative factor is $Q_{\infty}$ increased?
$Q_{\infty}$ (for $2k$) = $Q_{\infty}$ (for $k$)

On the time to reach $\frac{1}{2}Q_{\infty}$?
$t$ (to $\frac12 Q_{\infty}$), for $2k$) = $t$ (to $\frac12 Q_{\infty}$), for $k$)