Answer
The car is inside the garage with both doors shut for a time of $~~1.36\times 10^{-8}~s$
Work Step by Step
In part (a), we found that the length of the car according to Garageman is $1.93~m$
After the rear bumper reaches the front of the garage, the distance from the front bumper to the back of the garage is $6.00~m-1.93~m$ which is $4.07~m$
We can find the time for the car to travel this distance according to Garageman:
$t_{g2} = \frac{4.07~m}{(0.998)(3.0\times 10^8~m/s)}$
$t_{g2} = 1.36\times 10^{-8}~s$
Since the front door of the garage closes at time $t_{g1} = 0$, the car is inside the garage with both doors shut for a time of $~~1.36\times 10^{-8}~s$
