Answer
${\bf 25}\;\rm ns$
Work Step by Step
To determine the shortest duration radio-frequency pulse that can be amplified without distortion, we use
$$ \Delta f \approx \dfrac{1}{\Delta t } \tag 1$$
where $ \Delta f $ is the bandwidth, and $ \Delta t$ is the shortest pulse duration.
Noting that the bandwidth $ \Delta f$ is the difference between these two frequencies:
$$ \Delta f = 120 - 80 = \bf 40 \; \rm{MHz} \tag 2$$
Solving (1) for $\Delta t$;
$$\Delta t\approx\dfrac{1}{\Delta f}\tag 3$$
Plug into (2) into (3) to calculate the shortest pulse duration;
$$ \Delta t = \frac{1}{40\times 10^6} =\color{red}{\bf 25}\;\rm ns$$
Thus, the shortest duration radio-frequency pulse that can be amplified without distortion is 25 ns.
