Puzzle: a little boat problem
May 22, 2007 pm31 2:08 pm
A friend asks:
On a lake, with a very straight shore, you are 6 miles out in a small boat. The boat can make 6mph, but once you reach land, you need to get to a friend who is 2 miles up the shore. You can walk 2 mph. (in other words, the perpendicular boat-shore and the shore itself create a right triangle). What’s the least amount of time it would take you to reach your friend?
JD2718 
Looks like it needs calculus, but actually doesn’t: since you travel faster by boat, you’d have to be crazy to head anywhere but straight towards your friend, since landing anywhere else on the shore increases your total distance (by the triangle inequality) and decreases your average speed (since you walk slower than you boat). Thus, we just need to slap on an application of the Pythagorean theorem, giving us a distance of
and divide by 6 (our speed) to get the answer, 
, just a tad larger than one hour.
If we did want to use calculus, we’d have to start walking faster (or boating slower) — suppose our boat has a weak engine, only capable of 2 mph, but we’re willing to jog up the shoreline at 6mph. Then what’s our best time?
Totally unrelated: do your students take the NYCIML contest, and if so, in which division?
This year we only did the AMC. Last year we had AIME qualifiers, this year none. The good news was that the number of sophomores and freshmen sitting for the test was large. Over 10% of the school sat, and almost 20% in the lower grades.
In fact, our juniors (to be seniors) are competitive in an all-around sort of way, which has made them, generally, not serious math teamers. On the other hand, in the lower grades we have kids who may like this as a niche. I expect more a year or two down the road.
I was wondering if this was a trick question in some way, since shortest distance route (heading straight for your friend) is also the one where you’re traveling with the highest average speed (always 6mph)…
— Rachel
Nice and usefull post, thanks, this is one for my bookmarks!