$2\mathrm{.}23$ Run$-$Bike$-$Fun

A$$Run$-$Bike$-$Fun$$ sports event takes place every year in a small university town in Germany. Each participating team consists of two people. Both people have to complete a $15$ km course through a combination of running and cycling. Each team has one bicycle. Only one person is allowed to ride the bicycle at any one time, but team members can switch between running and cycling as often as they wish. The first team with both partners at the finish line wins.

At the beginning of the race, one person starts riding the bicycle, and the other starts running. After some time, the cyclist gets off the bicycle, puts it down, and starts running.When the other runner reaches the bicycle, he$/$she picks it up and starts cycling.

What is the optimal switching strategy? At which locations along the course should the switch$($es$)$ occur?

You may wish to begin by assuming that it takes no time to get on$/$off the bicycle,that both team members are x times faster at cycling compared to running, and that people run$/$cycle with constant velocity. Based on your own experience, estimate the value of x$.$When$/$where should you switch?

In reality, people get tired. How might you describe that? Would you use the same description for running and cycling? How does this affect the optimal strategy? Also,switching between cycling and running takes time. How does this affect the optimal strategy?What if two people with different abilities form a team?