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Extreme Value Problem

Problem setting: A man with a boat at point S at sea wants to get to point Q inland. Point S is d1 from the closest point P on the shore, point Q is d2 from the closest point T on the shore. The points P and T are at a distance of d from each other.


Question: If the man rows with a speed of v_r and walks with a speed of v_w at what point R should he beach the boat in order to get from point S to point Q in the least possible time?


Click here to play with or make your own GeoGebra simulation!
No calculus required!


The general case is actually a fairly difficult problem to solve mathematically!

It requires the numeric solution (using a graphing calculator or Newton's method) to find the root of a rational function or of a fourth-order polynomial. A sample problem with both solution methods is given here. A GeoGebra interactivity showing concerns about various numerical methods is given here.


Two subcases - good AP calculus problems (no graphing calculators or numerical methods).


Subcase 1: The point Q is on the shoreline (d2=0).


This problem is solvable (without any kind of calculator), it points out several interesting aspects of extreme values .


Click here for several different ways to pose and solve this problem.


Subcase 2: The man rows and walks at the same rate (speed): v_r=v_w .


This problem is solvable with calculus - it is harder than subcase 1. The cool thing here is that you can also solve this problem by noticing that since the rates are the same, the shortest path is the straight line joining S and Q (why?) and then you can solve this problem using only similar triangles.
(LFS: Since I am a BIG fan of multiple solutions, this is my favorite.)


Click here for complete solution to sample problem below.

View Sample Problem

Sample Problem  
A person on a boat in a lake is 4 km from the shore and must go to a point 8 km down the shoreline and 2 km directly inland from there in the shortest possible time. The person walks and the boat travels at the same rate of 6 km per hour.
a) Sketch the problem situation. Find total time t . This is a function of what variable?
b) Using calculus, find the point on the shore where the person should land to minimize the time.
c) Draw a scale drawing of the problem situation. What do you notice about the line connecting the start and stop points? Why do you think it is a straight line? Thinking hints: Suppose the person walks and the boat travels at the same rate of 10 km per hour. What changes and what doesn't change in the above? What do you notice about the derivative function? What would change if the person walked and the boat traveled at different speeds? Does this look like a harder problem?
d) Using only geometry (no calculus) and traveling on a straight line between the start and stop points, find the point on the shore where the person should land. (Your answer should be the same as in b.)

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Page last modified on January 23, 2010, at 12:19 AM