Sorry if this is posted in the wrong esction. I just couldnt find a better place.

[Question]

You and two friends are at standing at the beginning of a lake.

(Means that there is a totall of 3 persons)

On the opposite side lies a hut which you would like to visit.

The lake is shaped as a circle with the radius 5km.

There is a boat which holds maximum 2 persons.

The speed of the boat is 10m/s

Walking distance is 4m/s

What is the shortest time you could use to get to the hut with your

friends?

This is not a homework question, just a riddle i found. You do not have to

belive me, or give me the answer, just point me in the right direction.

[Difficulty]

To figure out the optimal intersection point between the person walking and

the person traveling by boat.

[Thoughts]

I think i figured out how to do parts of the problem.

Person A and B travels across the lake by boat. At the same time person C

starts walking around it. When Person A and B gets to the hut. Person A

travels back by boat to get person A.

I was thinking about many clever ways to find the optimal place between A

and B. And thought that it might be best by setting up two Parametric

equations. One for the walking person and one for the boat.

So here is how far I've come.

Radius = 5km

Diameter = 10km

$\displaystyle t=\frac{s}{v}$

$\displaystyle t=10*1000/10 $

$\displaystyle t=1000sec$

So it takes 1000sec to get across the lake.

$\displaystyle s=vt$

$\displaystyle s=4*1000$

$\displaystyle s=4000m$

So the person walking have walked 4000meters, when A and B are at the hut.

Now I tried to split the circle in half, and flip it. So that it looks

like a curve. With the lenght 10km and the height 5km.

But really I dont know how to find the optimal intersection point between

the boat and the person walking.

Bellow I tried solving it with parametric equations, but did not get very

far.

Parametric Equation for the boat. Look at the circle from the side...

$\displaystyle x:-10t + 10000$

$\displaystyle y:sin(a)*t$

Where (a) is the angle the boat have to aim to optimally hit the person

walking.

Equation for the person walking

circumference is given by $\displaystyle 2*pi*r$

The person must walk half this distance if there is no boat

$\displaystyle pi*r \;= \;pi5000\; \aprrox \;=\; 15708m$

The walking speed is 4m/s

$\displaystyle t=\frac{s}{v}$

$\displaystyle t=\frac{\pi*5000}{4}$

$\displaystyle t\;=\;pi*1250\; \approx \;3927m$

Which means that if you put $\displaystyle pi*1250$ into the eqation you should get

out 5000.

No idea how to set up the eqation for y though. I know the height is

reached after half the walking time or $\displaystyle pi*625$. So ploting in pi*625 should

give 5000 in the equation.

$\displaystyle y_1$ = 0 and$\displaystyle y_2 = 10km $

$\displaystyle x=4*pi*t$

$\displaystyle y=t(t-10000)$

Obviously wrong

So no idea if this parametric equation stuff is correct, or how to proceed.

I am not very advanced in my math, so if someone took it step by step i

would be very pleased