Excited by the recent warm weather Jill and Kelly are doing spring cleaning at their apartment. Jill takes an exponentially distributed amount of time with mean 30 minutes to clean the kitchen. Kelly takes an exponentially distributed amount of time with mean 40 minutes to clean the bath room. The first one to complete their task will go outside and start raking leaves, a task that takes an exponentially distributed amount of time with a mean of one hour. When the second person is done inside, they will help the other and raking will be done at rate 2. (Of course the other person may already be done raking in which case the chores are done.) What is the expected time until the chores are all done?
My current though process is to define the Kitchen ~ exp(1/30), Bathroom ~ exp(1/40), and Rake1 ~ exp(1/60) / Rake 2 ~ exp(1/30). From here, we can see that if cleaning the bathroom or kitchen takes longer than cleaning the other room + raking then obviously the expected time is just the expected time to clean that room. Outside of that, we can calculate the E(max(Kitchen,Bathroom)) = 30 + 40 - 1/(30 + 40). But after this I am lost as to how to get the final result.