I agree that you cut out two wedges, but I took that into account. Theta is the angle between the ray OA and the x-axis, so each wedge is this size, theta. Thus the total number of radians you must traverse around the circle is 2pi - 2theta.
:smack:
Let me claim again that this problem can not be solved as a math problem.
The minimum distance necessary to check every point on a circle, where a line may be tangent to the circle is the circumference, 6280K and we start 1000K from the circumference. That’s more than the fuel remaining.
Please prove that we must travel the entire circumference.
You can intersect a straight line by moving in a straight line, in BOTH directions, but this assumes you are not moving parrallel to the line. And the point of intersection could be far beyond ANY finite amount of fuel.
You must check out every point of tangency. The absolute minimun distance to do this is to travel the circle. Any larger circle or polygon would work too but would take even more fuel.
OK, I think I’m following the logic here. Spiff drives due east (or along the X-axis) to some point past the imaginary circle. (Point D in Viking’s notation.) He then turns left and follows the tangent line he is on until it intersects the circle. (Point A according to Viking.) Then he follows the circumference until he finds his ship. He will have to drive at most to point B to guarantee finding his ship … unless of course his ship is on the short arc from A to B, in which case he would have hit the coastline before D and followed it to his ship. All right then, if we call his origin O, then the angle AOD is arcsec(D). (Boy, it took a lot of thought to scrape that out of my memory, but I think it’s right.) DOB will be the negative of this angle. The distance from point D to point A is r * tan(theta). So the formula for the distance travelled is:
D + 1000 * (tan(theta) + 2pi - 2theta)
I’m way past the point where I’m able differentiate that mess, but by brute force I come to Achernar’s conclusion, i.e. D = 154.7, DIST = 6986.04 is the best Spiff can do by that strategy.
Hey, at least we showed aahala that it’s possible to beat r + 2pir. OK Spiff is possibly dead, but I still count that as a moral victory for us.
This is too easy.
Shoreline sand dunes form parallel to the shoreline. The shoreline in this problem is straight for thousands of kilometers.
Spiff should drive for 1000 km exactly perpendicular to the sand dunes. If he doesn’t find his space ship, he should drive in the opposite direction perpendicular to the sand dunes for 2000 km.
There’s his ship. Maximum distance traveled is 3000 km.
:smack:
I claim temporary insanity. My answer obviously only works if the original trip was directly away from the shoreline.
Although the problem doesn’t state it explicitly, surely Spiff knows the compass orientation of the shoreline (e.g., sea to the north, land to the south). Of course if Spiff knows the orientation of the shoreline he wouldn’t need to rely on the direction of the dunes to point him in the correct starting direction.
If Spiff knows which direction the shoreline is then I think he could get back to his ship with his remaining fuel. If he only has the orientation of the sand dunes to narrow down the shoreline direction to two opposite directions, then I’m not sure that he has enough fuel to find his ship. Perhaps someone more mathematically inclined can figure that out.
Unless he takes the exact same path back as he took out, irregularities in the terrain, i.e. dunes, could still make it impossible for Spiff to find his ship with the fuel available. This being a math problem, a single infinitely tall dune in the wrong place could stymy his plan for returning to the radius of the circle, or for finding the seashore.
I seem to recall seeing this problem somewhere. I think the solution is some sort of spiral, perhaps an exponential spiral. I seem to recall seeing the puzzle in terms of being on the ocean and there being a straight shore, or being a forest with a straight edge on one side. I don’t recall whether or not the distance to the shore was known or whether there was a “fuel constraint”, but just that a minimum-distance path to search for the straight edge was computable.
Wish I could help beyond this, but I just found this thread, and I’ll have to go off and play with it for a while. I doubt I’ll be able to prove it to be a minimum amount.
What? If you want to add a silly constraint like that, then there is no method that will work reliably.
See my first post in this thread for a solution which works if you know the orientation of the coastline (even if you didn’t travel back on the exact same path you originally took).
Exactly my point. The OP specifically mentions sand dunes, but nothing about their size distribution. Traveling 1000 km over dunes will not get you to the same point as traveling “as the crow flies.” Even an error of a km or two is likely to cause Pirx to miss his ship.
Well, then I guess that must be the answer! He can’t win!
Oh wait, I guess there is one possibility: that it’s a contrived scenario set up for the purpose of a math problem, and you should treat it as such. If that’s too much to ask, don’t bother doing the problem. It’s just for fun.
Besides, the problem doesn’t state that his odometer is based on the linear distance traveled at the wheels. I heard he has a super-high-precision accelerometer-based odometer which keeps track of acceleration (and thus relative position) in threespace, and then calculates distance traveled across the plane perpendicular to the force of gravity. So there.
I think that the problem may not be stated properly. I have seen a similar problem where you’re trapped in a forest and you know there’s a straight road that’s 1000 k from your position. But in that question, you just have to find the road, not a point that’s on the road and 1000k from your starting point. In that case:
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Call Spiff’s starting position the origin, and point your ship 60 degrees to the x-axis. Drive straight to the point ( 1000/sqrt(3), 1000 ). This is a distance of 2000/sqrt(3) (ie ~1000+154, as mentioned above).
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Take the tangent to the south-east, intersecting the circle at ( 1000*sqrt(3)/2, 500 ). This is a distance of 1000/sqrt(3).
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Drive 7/12 of the way around the circle, through (1000,0), (0,-1000), and stop at (-1000,0). This is a distance of 10007/6pi.
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Now you can just drive straight North to (-1000,1000). Distance: 1000. This step doesn’t work for our phrasing of the problem, because we can find the coastline, but not necessarily the ship.
So, total distance: 1000sqrt(3) + 10007/6*pi + 1000 = 6397 k.
However, that doesn’t work for this phrasing of the problem. And since the answer works out to so close to 6400, I think it is the intended idea.