How long is the drug effective?

Variations. $f'(t)=4e^{-t/2}\left(1-\frac t2\right)$, positive for $t<2$ and negative for $t>2$. So $f$ increases on $[0,2]$, then decreases on $[2,+\infty)$. The maximum is $f(2)=\frac8e\approx2.94>1$. Also $f(0)=0$ and $f(t)\to0$ as $t\to+\infty$.

Two crossings.
- On $[0,2]$, $f$ is continuous and strictly increasing from $0$ to $2.94$: it takes the value $1$ exactly once, at some $t_1$.
- On $[2,+\infty)$, $f$ is continuous and strictly decreasing from $2.94$ to $0$: it takes the value $1$ exactly once, at some $t_2$.

Hence $f(t)\ge1$ exactly for $t\in[t_1,t_2]$.

Locating $t_1$. $f(0.25)=e^{-0.125}\approx0.88<1$ and $f(0.3)=1.2e^{-0.15}\approx1.03>1$, so $t_1\in(0.25,0.3)$, i.e. $t_1\approx0.3$.

Locating $t_2$. $f(6)=24e^{-3}\approx1.19>1$ and $f(7)=28e^{-3.5}\approx0.85<1$. Refining, $t_2\approx6.5$.

Answer. The drug is effective from about $0.3$ h (18 minutes) to about $6.5$ h after intake, a window of about $6.2$ hours.