Simplest would be to get the density function for Zach arriving and bothering to stick around because Fatima is already there. At any moment between x=0 and x=10 the probability of that is e^(-x) (Zach arriving) times x^2/100 (Fatima already there). During the following 20 minutes, the second of these is one, instead. So evaluate the two definite integrals.
Or form double integrals, e.g.,
... where Fatima arriving first is the region under the line y=x.