Answer
The time between these two events is $~~4.18\times 10^{-7}~s$
Work Step by Step
We can find $\gamma$:
$\gamma = \frac{1}{\sqrt{1-\beta^2}}$
$\gamma = \frac{1}{\sqrt{1-0.900^2}}$
$\gamma = 2.294$
We can find the length of the tunnel in the train view:
$L = \frac{L_0}{\gamma}$
$L = \frac{200~m}{2.294}$
$L = 87.2~m$
In the train view, the length of the train is $200~m$
When the front of the train passes the far end of the tunnel (FF), the rear of the train has still not reached the near end of the tunnel (RN)
The rear of the train still needs to travel a distance of $112.8~m$
We can find the time this takes:
$t = \frac{112.8~m}{(0.900)(3.0\times 10^8~m/s)}$
$t = 4.18\times 10^{-7}~s$
The time between these two events is $~~4.18\times 10^{-7}~s$
