TEACHER NOTES

ISSUES RELATED TO THE PROBLEM

Problem 1 -- This is a nice problem to give at any time in a differential calculus class. The first problem is easily solved without calculus. In fact, after teaching the students optimization via the derivative, you will be guiding the students down the wrong path for solving this problem. If given before derivatives are introduced, the problem is easily solved upon realizing that if the river were thought of as infinitely thin, we could imagine the car being on the other side of the river, and so the shortest distance is a straight line path between the car and the fire with a brief stop at the river.

Other students may realize that the angle of reflection is the same as the angle of incidence. Finally, students should also notice that the triangles formed by the path taken are similar when the path distance is minimized. If given early in the course, the emphasis should be on open mindedness and problem solving skills. The students should be assured that the problem is easily solved with basic concepts learned in their high school geometry class. If given later in the course, this problem offers a nice example for the students to remember that calculus is not always the only or best route to optimization.

Problem 2 -- In this problem time is minimized. This may be somewhat confusing to some students since time is more often than not the independent variable, but in this case it is best to treat it as the dependent variable to be minimized as a function of the position where you hit the river bank.

Prerequisites

Problem 1 -- Triangle geometry, Pythagorean Theorem, reflection properties (not necessary, but helpful)

Problem 2 -- Optimization problems using the derivative

Time allotment - time management

This problem fills a 45-50 minute class period nicely. Have the students work in groups, and leave the last 10-15 minutes of class for groups to report on their varying attacks at the solution.

Expectations

Some students/groups will find this to be a quick and easy problem, while other groups may stumble endlessly without progressing at all. Students who have just been taught max/min problems will have little trouble approaching the problem using differential calculus as long they have a machine to aide with the calculations. (Many students will give up in frustration given a time constraint if faced with the algebra involved in this solution without a computer algebra system.)

Future payoffs

This problem emphasizes the need to keep a varied background, and that the most familiar tool isn't always the best one to use.

Extensions

(1) Determine the speeds s1 and s2 respectively that you must walk from the car to the river, and then the river to the campfire, so that the fastest way to do so at the prescribed speeds results in you hitting the river directly in the middle of the car and campfire (at x = 600 feet).

(2) Determine the speeds s1 and s2 respectively that you must walk from the car to the river, and then the river to the campfire, so that the fastest way to do so at the prescribed speeds results in you hitting the river at x = A feet.

(3) Repeat any of the earlier problems with the distances 300, 1200, and 600 changed to a, b, and c. What conditions must be placed on the parameters for the problem to have meaningful solutions? Study the effects of varying the parameters and write an essay on your results.

References and Sources