Optimal time for three people to travel

Question

Three men need to travel 60 km. They have a motorbike that can travel at 50 km/h but only two people can fit on it. They can walk at a speed of 5km/h. Can they get to their destination in 3 hours?

I found that they could do it in less than 3 hours but what is theoretically the best time they can do it in?

If anyone wants the explantion for how I found that they can do it in three hours, please comment and I will provide it.

Answer 1

Matti P has given a good approach in a comment. A and B travel most of the way, drop off B to walk the rest, C starts walking, A goes back to get C and they all arrive together.

Once you have a route that is described like this, you can parameterize it to find the minimum time it takes. Here you let $t$ be the time that A drops off B and $u$ the time that A picks up C. B arrives at $t+(60-50t)/5$. C arrives at $u+(60-5u)/60$. A arrives at $t+(50t-5u)/50+(60-5u)/50$. You want all of these to be the same, which lets you solve for $t,u$ and find the total time.

The challenge is often to find all the reasonable approaches. Here one might reason that the bike carries two people, so each person should walk $\frac {50}3$ km and ride $\frac {100}3$ km. That fails because the walk takes more than $3$ hours. If you didn't think of the other approach you might say the problem cannot be solved.

Answer 2

Suppose B rides the motorbike and A and C are passenger/pedestrians. Let the journey start at O and end at D. It is clear that B has to do separate journeys for the others, who can also spend time walking, and that they don't necessarily need to arrive at D all at the same time, as long as they are all there within 3 hours. We assume all drop-offs and changeovers are instantaneous.

We can break the whole episode into five time intervals:

1. B takes A to a point P so that OP $=x$ while C starts to walk.
2. B returns to pick up C while A is left to continue walking.
3. B and C go all the way to D.
4. B returns to pick up A.
5. A and B go to D, and the process is complete.

Accordingly,

1. $t_1=\frac{x}{50}$
2. In this time, C has walked $5t_1=\frac{x}{10}$. B and C are approaching each other at speed $55$, so B reaches C in time $t_2=\frac{9x}{550}$. In this time, both A and C have walked a distance $\frac{9x}{110}$.
3. B and C go all the way to D which is a distance $60-x-\frac{9x}{110}=60-\frac{2x}{11}$. This takes time $$t_3=\frac{1}{50}[60-\frac{2x}{11}]=\frac 65-\frac{2x}{550}.$$
4. B returns to pick up A who meanwhile in time $t_3$ has walked a distance $5t_{3}=6-\frac{2x}{110}$. Therefore, when B sets off from D to pick up A, they are separated by a distance $$60-x-\frac{9x}{550}-6+\frac{2x}{550}=54-\frac{557x}{550}.$$ They are approaching each other at speed $55$ so they meet in time $$t_4=\frac{1}{55}[54-\frac{557x}{550}].$$
5. While B was coming back to pick up A, A had walked an extra distance $5t_4=\frac{1}{11}[54-\frac{557x}{550}]$, so the remaining distance between A and B is $$54-\frac{557x}{550}-\frac{1}{11}[54-\frac{557x}{550}]=\frac{540}{11}-\frac{557x}{605}.$$ They drive this remaining distance in time $$t_5=\frac{1}{50}[\frac{540}{11}-\frac{557x}{605}].$$

Putting all this together, we get a total time of $$\frac{174}{55}-\frac{62x}{15125}$$

This is less than $3$ provided $$x>39\frac{57}{62}$$

So the optimal time is when $x=60$, that is, B drives CA all the way to D and comes back to pick up C. The journey time is 2 hours 55 minutes 3.67 sec