Answer
$5.0\;\rm s$
Work Step by Step
We have here a damped oscillation system. We know that the amplitude of these kinds of systems is given by
$$A=A_0e^{-t/2\tau}$$
The author told us that at $t=10$ s, the amplitude is $A=0.368A_0$,
so that
$$0.368 \color{red}{\bf\not}A_0= \color{red}{\bf\not}A_0e^{-10/2\tau}$$
Hence,
$$\ln(0.368 ) =\dfrac{-10}{2\tau}$$
Solving for the time constant $\tau$;
$$\tau =\dfrac{-10}{2 \ln(0.368 )}$$
$$\tau =\color{red}{\bf 5.0}\;\rm s$$
