The independent rates add so you want to solve
$CDF[Poisson[(3+2)t],400]=0.9$
You can use the Normal approximation to the Poisson distribution to get a reasonably accurate answer.
$Poisson[\lambda] \text{ is approximately modelled by }Normal[\lambda,\sqrt{\lambda}]$
As a ballpark guess you would expect 400 cars passing with a rate of 5 cars/minute to pass in approximately $\dfrac{400}{5} = 80~min$
I get 75.1 minutes as an answer. Is the 85.1 a typo?
Thank you!
The solution written in the book is 85.1 min.
I would like to ask how you come up to 75.1 minutes? Also P(T_400<=a)=0.90, isn't this an exponential distribution since it's about the distribution of the time 400 cars have passed?
I appreciate the help.