Boat in the fog

Geometry Level 1

You are on a boat on the sea and the weather is so foggy that you cannot see anything beyond your own boat. You have a GPS, so that you can plan and accurately execute a travel path, but you do not have a map. You have fuel for 7 miles of travel.

You know that the shore is a straight line, exactly 1 mile from you. You do not know what is the direction of the shoreline. Is there a travel plan that will certainly lead you to the shore before the fuel runs out?

No Yes Only if you know the direction to the shore There is not enough information to answer

2 solutions

Laszlo Mihaly
Jul 21, 2017

The simplest way to make it to the shore is to take a straight path in one (random) direction, and make a circle of radius 1 mile around the original position. In the worst case scenario this takes 1+2 \pi = 7.28 miles, so it does not work.

A shorter path can be constructed as this: 1 mile straight, followed by a 3/4 of a circle and one mile straight along the tangent of the circle. In the worst case scenario this takes 1+(3/2) \pi + 1 = 6.71miles.

Interesting question! Can we optimize this further?

Calvin Lin Staff - 3 years, 10 months ago

Graham Vdb
Oct 13, 2017

The easiest way to approach this problem is through trial and error. Since you have a GPS, no matter where you go you can always find your way back to your origin point.

So to solve this problem you could go north for a mile, if the shore was there then great, if not then return to your origin point. Worst case this trip would cost you 2 miles worth of fuel. Do this for South and East. If you don't find shore then you will be at your origin point with 1 miles worth of fuel left. Since you have exhausted all your possibilities you can be assured that the shore is to your West and you can use your last miles worth of fuel going left and finding shore.

Sorry, this does not work. You assumed that the shore is either parallel to North-South or it is parallel to East-West. For your method, the worst case scenario happens when the shore makes a $45^{\circ}$ angle to the North-South. Therefore you must go $2 \times \sqrt{2}= 2.82$ miles to North to be sure that you do not miss it. Similarly you have to do the same in the other 3 directions. Sure enough you will get it in one of the tries, but in the worst case you will have to travel $3 \times 2.18 + 1.41=9.9$ miles.

Laszlo Mihaly - 3 years, 8 months ago

If the shore is a straight line then simply go one mile north then proceed one mile east then two miles south and finally two miles west, worst case scenario or 6 miles total. In essence imagine your location is at the center of a two by two square.

Greg Grapsas - 2 years, 11 months ago

You did not include the last side of the square, going 2 miles north. Without that you miss the shore if it is in the north-west direction from you. Including the extra 2 miles makes the trip a total of 8 miles.

Laszlo Mihaly - 2 years, 11 months ago

Boat in the fog

2 solutions

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