I HATE RELATED RATES

Here is the link to this weeks Car Talk Puzzler: "Row, Row, Row, Your Boat"


Here is what I have so far: 


(note: I got no where, which was not surprising.):

From Random Graphic Images

Here is the SHORT VERSION of the puzzle: You put your boat in and row up river for one mile.  Your hat flies off.  You continue rowing for 10 minutes then decide to go back for your hat.  You catch up to the hat at the same place you started rowing.  You row at the same pace going down river as you did going up river.


My cheat was to 'google-search-hat-floats-downstream' for how to do related-rates, (and a big hint for this particular one, hopefully) and apply it to this problem.  I wind up doing this about once a year and then because the mind tends to forget painful memories I forget how to do related rates until the next time I have to do one.


Of course since this is the 'Car Talk Puzzler' then there must be some nice way to conceptualize this puzzle that does not require a lot of algebra.


And there is and here it is: What It Means To "Understand" Something!. (This was one of the 'let-me-google-that-for-you' hits from above.  


The whole article is well worth reading, in its entirety, but for the purposes of this post the bit about the hat floating down stream is of most concern here.


A key to understanding this problem (as pointed out in the link above) is to not think about rowing against the current or rowing with the current but instead think of walking along one of those 'glide-ways' that they have in airports. 


You are walking along the glide way and you drop your hat and continue to walk for however long until you decide to turn around and walk back and pick up your hat.  It will take exactly the same amount of time to walk away from the hat as it will to walk back to the hat.  So if you decided to turn around after 15 seconds then it will take you 15 seconds to walk back for your hat. (another way to say this is to say that: THE HAT DID NOT GO ANYWHERE WITH RESPECT TO THE SURFACE OF THE GLIDE WAY YOU WALKED AWAY FROM IT (on the surface of the glide way) THEN WALKED BACK TO IT).


Notice, however one of three cases applies:


[CASE-A] If you pay attention to the art work on the walls then you will notice when you are walking the 'correct' way along the slide way if things go by faster than normal walking.


[CASE-B] Or you will notice that you are walking the 'wrong' way along the slide way if things go by slower than normal walking.


[CASE-C] Or if the slide way is stopped then you will notice that the art work on the walls changes as if you were not even ON the slide way, things will going by at normal walking speed.  


However, and this is the punch line, in all three cases it will take the SAME amount of time to go back and pickup your hat as you spent walking away from it, because the hat just lay there on the slide way. 


NOTE: This does assume if you are walking the 'wrong' way on a moving slide way that the hat does not arrive at the end before you get back to it.  If it does then you would catch up to it more quickly than otherwise. 


The reason the time to get back to your hat is the same in all three cases is that your motion with respect to the surface of the slide way is due ONLY to your walking motion.  


If you walk the 'correct' way then it is as if you are paddling 'downstream-with-the-current'.  


If you walk the 'wrong' way then it is as if you are paddling 'upstream-against-the-current'.  


If the glide way is stopped, or you are not walking on the glide way it is as if you are paddling 'in-a-zero-current-stream'. 


Still another way to say things is to say: You and your hat are NOT being carried along the glide way, the glide way itself is the thing that is moving and you and your hat are being carried along (or not) the 'bank' of the glide way.


So in terms of the original puzzle:  


How fast was the current if you notice; when you grab your hat out of the water, (which we now understand will take exactly 10 minutes to get to) you and the hat are right back where you started (the dock)? 


And you also notice this an interesting coincidence, that you grab your hat right at the dock, and probably think that this coincidence allows you to actually compute the current rate. Well, really, if you just grabbed your hat and noted where you were you were you would probably say, "Man, the current must be really fast for me to travel all this way in 10 minutes!" 


Now here is the math.


The rate of change of the distance between you and the 'DOCK' when traveling 'upstream' is r-c (where r is the rate you row, and c is the rate of the current).  This is because you are rowing against the current and it is slowing you down, relative to the bank.


The rate of change of the distance between you and the 'starting-point' when traveling 'downstream' in the current is r+c. Likewise this is because you are rowing with the current and it is speeding you up, relative to the bank.


Assume everything is miles per hour so (10 minutes is written as 1/6 of an hour).  


Also notice that when miles per hour is multiplied by time (hours) then distance results (in other words (m/h)*h = m. (And miles per hour means miles divided by hours). 


So now:  


If d = distance traveled along the bank in the 10 minutes after loosing your hat. 
then d = (r-c)/6.


And the total distance traveled from the starting point before turning around is 1+(r-c)/6. Because we traveled one mile from the dock before our hat flew off.


And also:


We have the case that it takes 10 minutes (1/6 hour) when traveling at r+c to travel all the way back to the starting point which is 1+(r-c)/6 miles away from where we turned around (as we figured out above). 


Also notice that when distance traveled (miles) is divided by time (hours) then miles per hour results: the formula is m/h.


So we now get the following formula for miles per hour down stream when we divide total distance paddled down stream by total time:


[1+(r-c)/6]/(1/6) = 6 + r-c 


That formula is the speed traveling downstream.


However:  From just thinking about traveling downstream we already know the speed is just the rowing rate plus the current rate.  The formula is r+c.


Setting these two formulas equal to each other we get:


6 + (r-c) = r+c
    6 - c = c
        6 = 2c
        3 = c


And the answer to the question of whether or not the speed of the current can be computed is: Yes! And the rate of the current is 3 miles per hour.