Expert: Paul Klarreich - 4/30/2006

Question
Dear Paul,

my name is Julia and I have a question to your answer to the following published question:

Expert: Paul Klarreich
Date: 11/27/2005
Subject: Application of Differentation

Question
A woman at a Point A on the shore of a circular lake with radius 2 mi wants to arrive at point C diametricallly opposite A on the other side of the lake in the shortest possible time. She can walk at the rate of 4 mi/h and row a boat at 2 mi/h. how should she proceed?

I have been working on a similar problem (just other values). Even if this problem is an oldie, it's very interesting, but I am not very satisfied with the solution. So I was looking for similar problems and I found yours.
I think your answer is not complete. You stopped with x=1. But this does just mean, there is a critical point at x=1. Unfortunately, the second derivative at this point is negative, hence x=1 is a maximum which means the woman will arríve at the other side in the longest possible time.
As there is no other critical point between -2 and 2, the shortest possible time is for x=-2, which means, she is only walking.

So, my actual question is:
1) Is my solution correct, and if yes,
2) I find it unsatisfying that the final solution is "only walking". Do you know, if, with using other numbers (change the diameter of the lake or change the velocity) you can actually get a real "rowing-and-walking" solution, or will it always be either "only rowing" or "only running"?


Thanks for your thoughts,

Julia  

Answer
Hi, Julia,

You are indeed correct.  I must have been in too much of a hurry at the time, and totally neglected to check the endpoints or the second derivative.

It is indeed true that the value of  x = 1  provides not a minimum but a maximum, and the minimum occurs at the endpoint value x = -2, as you said.

So when you said: "I think your answer is not complete." you really meant to say:

"You blew it, old man.  You got lazy and forgot to check at the endpoints.  Don't be in such a hurry next time."

But of course your manners are too good for you to say anything like that.  Your mother brought you up well.  Thank you for checking up on me.

About your second question -- can the numbers be adjusted so that the minimum time does in fact occur somewhere inside the interval -- that may take a little thinking, so I will try to send a followup when I can.

Paul