| Nit: I assume .1 meant the boat could stay within d of both banks for
any d greater than or equal to 50 yards.
Here's an interesting river:
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx A xxxxxxxxxxxxxxxxxxxx
-----------------------------------------------------------+xx
> > > |xx
--------------------------+ +-+ |xx
xxxxxxxxxxxxxxxxxxxxxxxx B| |C| |xx
xxxxxxxxxxxxxxxxxxxxxxxx | S | | |xx
xxxxxxxxxxxxxxxxxxxxxxxxxx| |x| |xx
xxxxxxxxxxxxxxxxxxxxxxxxxx| |x| |xx
xxxxxxxxxxxxxxxxxxxxxxxxxx| |x| |xx
xxxxxxxxxxxxxxxxxxxxxxxxxx+---------------------------+x| |xx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx D xxxxxxxxxxxxxx| |xx
--------------------------------------------------------+ |xx
< < < |xx
-----------------------------------------------------------+xx
xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
X's represent land. Let's say point 'A' is 100 yards from points B and C.
How far from point A can point D be? I think it can be nearly 200 yards.
Thus, the swimmer S can be nearly 100 yards away from the nearest bank,
and the boat can't stay closer than about 100 yards from both banks.
You can "bump" point A farther away from D using the same technique: provide
another meander above A for it to be close to. Can you also move B and C
farther apart?
|
| The straight-forward answer of 50 yards is too, uh, ... straight-forwrad.
The problem is to find a convoluted river such that the banks are within
100 yards of each other, but the 100 yards is somtimes across solid ground.
Consider the following form of river:
+------------------------+
/ | B
+ +-----------------------+ |
| |wwwwwwwwwwwwwwwwwwwwwww| |
| |wwwwwwwwwwwwwwwwwwwwwww| |
| |wwwwwwwwwwwwwwwwwwwwwww| |
| |wwwwwwwwwwwwwwwwwwwwwww| |
| |wwwwwwwwwwwwwwwwwwwwwww| |
| +-----------------------+ +
A | /
+------------------------+
Here, the river starts narrowly from A (the '-'), goes north, then east,
then south to a very wide part of the river (the 'w') which goes west.
Then it narrows and goes south, then east, then north to b.
The banks are within 100 yards of each other (in the above diagram,
A and B are about 1000 yards apart).
> a. A boat sails down the river, trying to stay always within a
> distance d from both banks. For what values of d can we guarantee that
> such a trip is possible?
In the above example, d can be as much as 100 yards (less the width of
the boat). This width will always guarantee such a trip is possible,
because for any river, the boat can make its trip hugging either bank,
which by definition is within 100 yards of the other bank.
> b. How far can a swimmer in the river be from the nearest bank?
In the above example, the 'w' region can be arbitrarily large, so if
the swimmer is in the middle of it, there is no theoretical upper bound
on the swimmer's distance from a bank.
But perhaps the answer to a is larger, much larger; consider putting
town B on an island in the middle of a large lake. :^)
|