(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. 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Next, find the shortest travel time (and corresponding route) possible given that you can walk faster without water than with it. :[font = section; inactive; Cclosed; preserveAspect; startGroup] GENERAL INFORMATION :[font = subsection; inactive; preserveAspect; endGroup] FileName: CAMPFIRE Full title: Put out the camp fire Last Update: 5/28/96 Developers: Sandra K. Dawson, Glenbrook South High School, Glenview IL 60025 USA; Dave Horn, Roger High School, Michigan City IN 46360 USA; Aaron Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Contact: Aaron Klebanoff, Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute IN 47803 USA. Phone: 812-877-8151. Email: Aaron.Klebanoff@rose-hulman.edu. FAX: 812-877-3198. Support: The production of this material is supported by the National Science Foundation under Division of Undergraduate Education grant DUE-9352849: Development Site for Complex, Technology-Based Problems in Calculus with Applications in Science and Engineering and the Arvin Foundation of Columbus IN. :[font = section; inactive; Cclosed; preserveAspect; startGroup] STATEMENT OF PROBLEM :[font = subsection; inactive; preserveAspect] You are on a camping trip and your camp site is 300 feet from the river. You parked your car 1200 feet down the river from the campsite and 600 feet from the river bank. As you break camp, you need to get a bucket from the car and water from the river to put out the campfire. :[font = subsection; inactive; preserveAspect] 1. What path would give you the shortest distance as you walk from the car to the river and then from the river to the campfire? :[font = subsection; inactive; preserveAspect; endGroup] 2. Assume that you can walk at 3 feet per second from the car to the river but you slow down to 1.5 feet per second when you are walking with a bucket full of water. Find the path you would take to minimize your time. Is it the same path that minimizes your distance? :[font = section; inactive; Cclosed; preserveAspect; startGroup] KEYWORDS :[font = subsection; inactive; preserveAspect; endGroup] Distance, minimization/optimization, derivative. :[font = section; inactive; Cclosed; preserveAspect; startGroup] TEACHER NOTES :[font = subsection; inactive; preserveAspect; startGroup] ISSUES RELATED TO THE PROBLEM :[font = subsubsection; inactive; preserveAspect] 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. :[font = subsubsection; inactive; preserveAspect] 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. :[font = subsubsection; inactive; preserveAspect; endGroup] 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. :[font = subsection; inactive; preserveAspect; startGroup] Prerequisites :[font = subsubsection; inactive; preserveAspect] Problem 1 -- Triangle geometry, Pythagorean Theorem, reflection properties (not necessary, but helpful) :[font = subsubsection; inactive; preserveAspect; endGroup] Problem 2 -- Optimization problems using the derivative :[font = subsection; inactive; preserveAspect; startGroup] Time allotment - time management :[font = subsubsection; inactive; preserveAspect; endGroup] 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. :[font = subsection; inactive; preserveAspect; startGroup] Expectations :[font = subsubsection; inactive; preserveAspect; endGroup] 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.) :[font = subsection; inactive; preserveAspect; startGroup] Future payoffs :[font = subsubsection; inactive; preserveAspect; endGroup] This problem emphasizes the need to keep a varied background, and that the most familiar tool isn't always the best one to use. :[font = subsection; inactive; preserveAspect; startGroup] Extensions :[font = subsubsection; inactive; preserveAspect] (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). :[font = subsubsection; inactive; preserveAspect] (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. :[font = subsubsection; inactive; preserveAspect; endGroup] (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. :[font = subsection; inactive; preserveAspect; endGroup] References and Sources :[font = section; inactive; Cclosed; preserveAspect; startGroup] POSSIBLE SOLUTION(S) :[font = subsection; inactive; preserveAspect] 1. (Geometry Solution) Imagine that the campfire is on the opposite side of the river, which is infinitely thin. Then, it is clear that the shortest distance between the campfire and the car is a straight line path, and since this gets you to the river 400 feet from the fire, that must be the optimum place to aim for at the river bank. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 1. (Calculus Solution) :[font = subsubsection; inactive; preserveAspect; startGroup] Let the river bank be the x-axis with the origin placed directly below the fire. In other words, the campfire is at coordinates (0, 300) and the car is at coordinates (1200, 600). We are seeking the x-value which minimizes the path from the campfire to the river and back to the car. For sake of clarity, call the point (x, 0) where we pick up water from the river W. The distance from the campfire to W is Sqrt[x^2 + 90000] and the distance from W to the car is Sqrt[360000 + (1200-x)^2]. The following Mathematica code yields a formula for the total distance. :[font = input; preserveAspect; endGroup] distance[x_] = Sqrt[x^2 + 90000] + Sqrt[360000 + (1200-x)^2]; :[font = subsubsection; inactive; preserveAspect; startGroup] It's helpful to plot the distance function to see where it is minimized. :[font = input; preserveAspect; startGroup] Plot[distance[x], {x, 0, 1200}, PlotLabel -> "Distance D vs. Water Pick-up Location X", AxesLabel -> {"X [feet]", "D [feet]"}] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] Graphics["<<>>"] ;[o] -Graphics- :[font = subsubsection; inactive; preserveAspect; startGroup] In order to minimize the distance, we look for critical points by differentiating the distance function and setting it equal to zero. :[font = input; preserveAspect; startGroup] Solve[distance'[x]==0,x] :[font = output; output; inactive; preserveAspect; endGroup; endGroup] {{x -> 400}} ;[o] {{x -> 400}} :[font = subsubsection; inactive; preserveAspect; endGroup] Therefore, W is 400 feet down the river bank from the campfire or equivalently, 800 feet up the river bank from the car. :[font = subsection; inactive; Cclosed; preserveAspect; startGroup] 2. To minimize the time, we use the fact that T = D / R. :[font = input; preserveAspect] time[x_] = Sqrt[x^2 + 90000]/1.5 + Sqrt[360000 + (1200-x)^2]/3; :[font = input; preserveAspect; startGroup] sol = Solve[time'[x]==0,x] :[font = output; output; inactive; preserveAspect; endGroup] {{x -> 144.7865630391298}} ;[o] {{x -> 144.787}} :[font = input; preserveAspect; startGroup] time[x]/.sol[[1]] :[font = output; output; inactive; preserveAspect; endGroup] 626.6968603136011 ;[o] 626.697 :[font = input; preserveAspect; startGroup] time[x]/60/.sol[[1]] :[font = output; output; inactive; preserveAspect; endGroup] 10.44494767189335 ;[o] 10.4449 :[font = subsubsection; inactive; preserveAspect; endGroup; endGroup] Therefore, W is about 145 feet down the river bank from the car or equivalently, 1055 feet down the river bank. And the total time this will take is 627 seconds or 10.5 minutes. :[font = section; inactive; Cclosed; preserveAspect; startGroup] ISSUES IN SOLUTION :[font = subsection; inactive; preserveAspect] Some students will set up their axes differently. For this problem, that should cause no additional trouble. This is nice, since students can take pride in successfully completing the problem in a way that is likely to be different from their neighbors. To further this, some students might simplify the problem a bit by working in units of 100 feet. :[font = subsection; inactive; preserveAspect; endGroup; endGroup] The algebra involved in successfully minimizing the functions in this problem is quite involved if done by hand. ^*)