Lake Walk Optimization Problem

[chee]

A woman at a point A on the shore of a circular lake with a radius of 2 miles wants to arrive at the point C opposite A on the other side of the lake in the shortest possible time. She can walk at a rate of 4 miles an hour and row a boat at 2 miles an hour. How should she proceed?
http://scrllock.dyndns.org/28.jpg

 

[AKG]

She should row across in 2 hours. If the scenario were that her speed rowing and walking were the same, then it would be obvious that the most direct route would be best. That route would be rowing across. Now, with the added information that she rows faster than she walks, it is obvious that any other route would a) be longer, and b) require her to walk, which takes time. If you can choose a path that let's you go the shortest distance possible at the fastest speed possible, it's obviously the optimum path.

This problem seems a little too simple. Are you asking how to do this type of optimization problem in general? Basically, if she rows to a certain point on the perimeter on the lake, she will have to walk around the edge the rest of the way. You can determine how far she rows, and thus where she gets off on shore, and then in turn how far she walks, all starting with the angle at which she leaves shore (say the angle is zero degrees if she goes straight across rowing. Since you can express time in terms of one variable, you would differentiate time with respect to the angle, and find where the derivative is zero, giving you one critical point. You would also have to test the extreme values for the angles, namely zero degrees (rowing right across) or 90 degrees (walking the whole way). Plug the critical point and extreme points into the equation for time, and choose the minimum time. In this problem you've given, the answer is obvious right off the bat.

 

[HallsofIvy]

Apparently Chee edited this after AKG's response. Now the woman walks faster than she rows!