A cool problem...?

Question

You're on a raft on a body of water. You are exactly 10 miles from shore, but you don't know in which direction and there is a thick fog that won't let you see the shore until you touch it.

You are able to maneuver the raft with precision, i.e. you can travel in straight lines, arcs of circles, etc. and exact distances. Which path will give you the smallest maximum distance that you will have to travel in order to find the shore?

Debster has the right idea for a start... that would give you a maximum distance of 10 + 2*pi*10 = 72.8 miles.

But this can be improved on...

The shore is a straight line. Also, there are sharks.

bpiguy: the shore isn't confined to a square. What if you happen to be going parallel to the shore? You'll go a very long way without hitting ground...

Scott, you're getting close... You can still improve on this, the beginning of your trip is less than optimal.

bpiguy is closest, so I'll give him the best answer... but it still isn't optimal.

Answer 1

Go 10 sqrt 2 = 14.14 miles in one direction, and if you guessed wrong, turn around and go 20 sqrt 2 = 28.28 miles in the opposite direction. By that time, you'll hit shore no matter where the shoreline is. Maximum travel distance is 30 sqrt 2 = 42.43 miles.

To see this, draw a square circumscribing the circle. Travel to one corner of the square, then turn around and travel toward the opposite corner.

[Edit -- second (and last) try ... I saw your note, Benoit, so I'll try again ...

Go ten miles in any direction. Turn left or right and follow the circle of the way around. Then go straight, parallel to your original track, for another ten miles.

Total distance 10 + (3/4)(2 pi 10) + 10 = 20 + 15 pi = 67.12 miles. End edit.]

Answer 2

Statistically, it seems that on average, the shortest path would be to travel a straight line of 10 miles (i.e. the radius) in any direction, and then travel the arc of the circle until you reach the shore.

Edit: I think I figured it out. It depends on the assumptions you make. If you assume that the shore is a straight line and at least 20 miles long, I think you could solve it this way:

Go 10 miles in any direction. Then, take a 45 degree angle and keep going until you hit the point that would be on the circumference of the circle w/ radius 10. I believe that distance would be square root of 20. Then, take a 90 degree angle until you hit the circumference again. Do the same thing twice more until you end up making a diamond within the circle. So, when you do the math (not sure if I'm remembering my trig correctly), I think it's 10 + 4*Square root of 200 = 66.6.

Edit again: I don't think my 2nd stab is correct. I think bPiguy got it, though.

Answer 3

I know this isn't the right answer, but for a laugh:

The path that heads directly to the shore, does not pass go and does not collect £200. Giving you a maximum of 10 miles every time!! Would you want to row a raft 10 miles though? seriously?

Answer 4

Geometrically this means that you're at the center of a circle of radius 10 miles and want to get to the boundary. You'll need to travel straight in any direction before you know anything. If you don't hit shore

Needs more thought, and sketches.

Answer 5

Huddle down in the center of your raft and await clearing of the fog. When it clears sufficiently for you to see the shore, begin paddling in a straight line for the shore. Spend the idle time til the for clears with fishing, planning the menu for tomorrow's supper, and possibly playing with your toes.

Answer 6

If memory serves, a logarithmic spiral.

Answer 7

arcs of circles