2008 Excellence in Mathematics Contest Team Project B. School Name: Group Members:

Transcription

1 2008 Excellence in Mathematics Contest Team Project B School Name: Group Members:

2 Lassie - TEAM PROJECT B 2008 Excellence in Mathematics Contest The Team Project is a group activity in which the students are presented a series of mathematical problems relating to a specific theme. The team members are to solve the problems and write a narrative about the theme which answers all the mathematical questions posed. Teams are graded on accuracy of mathematical content, clarity of explanations, and creativity in their narrative. Many of you have probably seen the famous Lassie save the day by rescuing someone, alerting someone of imminent danger, or just providing the companionship in a way that only man s best friend can. But, does Lassie know Calculus? Let s be honest, Lassie probably does not know calculus and most likely cannot differentiate even a simple polynomial. However, could it be that Lassie is able to innately determine the optimal solution to a problem? There are many examples in nature where optimal solutions exist. Soap bubbles, for example, will always minimize surface area for a given volume. Bumblebees also minimize surface area by creating their beehives in such a way that the minimum amount of wax is needed. Birds and fish naturally choose migrating patterns in a way that minimizes the amount of energy expended during the migration. But, what about the canine species? Do they have the natural ability to optimize the time needed to travel from point A to point B? Let me explain. Imagine that Lassie happens upon a situation along a beach where a person, out in the ocean, struggles to keep his head above water. How should Lassie choose her path so that the time needed to reach the child is minimized (see below)? Should she immediately jump into the water and begin swimming toward the person since the shortest distance between two points is a straight line? Or, should she run down the beach until she is able to turn at a right angle and swim directly to the person? Or, should she run a portion of the way, then plunge into the ocean and swim diagonally to the drowning person? B x D y C z 2

3 Part I Let r represent the speed at which Lassie can run and s represent the speed at which Lassie can swim. Also, let T(y) represent the time it takes Lassie to get to the drowning person given that she jumps into the water at point D, which is y meters from point C. Find the value of y that minimizes T(y) and provide a complete interpretation of this result. That is, consider cases where r < s, r > s, and r = s. We used this diagram to create a function that relates the time it takes Lassie to get to the drowning person to the distance the drowning person is from Lassie. We were specifically interested in knowing how far Lassie should run down the beach before jumping in and swimming diagonally to the person. From this diagram, we created the following equation. z y x + y T( y) = + r s In this equation, T( y) represents the time necessary to reach the drowning person, r represents the rate at which Lassie runs, s represents the rate at which Lassie swims, and x, y, and z correspond to the distances shown in the diagram. Additionally, z y represents the distance Lassie runs along the beach, and x + y represents the distance Lassie swims in the water. We know that distance = rate*time, and therefore time = distance/rate. Thus to find the time it will take Lassie to reach the drowning person, we took the distance traveled and divided by the rate Lassie traveled for that distance. Therefore, the time it took Lassie to run along the beach is represented by z r y, and the time it took Lassie to swim in the water is represented by s To answer your question, we looked at a situation in which the distances z and x remained constant, as did Lassie s running and swimming rates. Therefore, the time it takes Lassie to reach the drowning person would only change as the distance y changed. In order to find the minimum time it will take for Lassie to reach the drowning person and how far down the beach Lassie should run before jumping into the water, we needed to take the derivative of our function and set it equal to zero. At a maximum or a minimum point on a function, the slope of the function, or the derivative, will equal zero. Therefore we can use the derivative to find the minimum point on our function, which is shown below. x + y y 1 T ( y) = s x + y r At this point we set the derivative equal to zero and solved for y. This gave us the equation shown below. sx y = s + r. This equation can be used to determine a value for y that will minimize the original time function, and therefore allow Lassie to reach her goal in shortest time possible. The equation shows that distance that Lassie should run along the beach depends on the vertical distance out in the water of the victim, as well as Lassie s running and swimming speeds. In order for this equation to work, it must be the case that the running rate is faster than the swimming rate. In the denominator of the equation we see s + r. When r is greater than s, we will end up with the square root of a positive number, thus giving us a positive value for the distance y. If however s was greater than r, we would end up with the square root of a negative number in the denominator. This would give us an imaginary number for the distance y, which would be very difficult for Lassie to measure while running! Additionally, if r and s are equal, we would end up with the square root of zero in the denominator. Division by zero is always a tricky thing, especially for dogs. Therefore, when it is the case that running speed is greater swimming speed, the above equation will allow you calculate the distance down the beach that Lassie should run in order to reach the victim in the minimum amount of time. 3

4 Part II Through experimentation, it has been found that dogs will always run down the beach, enter the water at some point D, and swim to the desired destination (such as a drowning person, or more likely, to retrieve a tennis ball). But, do dogs do this by choosing the optimal path that you found in Part I? One mathematician, Timothy Pennings (2003), investigate this situation with his dog, Elvis. By clocking Elvis while chasing a tennis ball on a beach and swimming, the following data was collected. Running and swimming times for Elvis Running times (in seconds) for 20 meters Swimming times (in seconds) for 10 meters Using this data and your results from Part I, determine the relationship between the distance y and x. To continue our analysis and truly determine whether Lassie knows calculus, we determined actual running and swimming rates to plug in for r and s in our equations. The values for r and s were obtained from the data of running and swimming times you sent in your letter. The average running and swimming time was determined. The distance for which the time was recorded was then divided by the average running or swimming time to give a rate in meters per second. The running rate (r) was found to be 6.35 m/s and the swimming rate (s) was determined to be m/s. This gave us the equation shown below y T ( y) = x + y As stated earlier, the point where the derivative of a function is equal to zero corresponds to a maximum or minimum on the original function. Therefore we set this equation equal to zero and used Maple to solve for y. The following equations show the result of this work. y= 0.139x This tells us that when the time is minimized, the relationship between distances y and x is modeled by the equation y = x. This means that Lassie should run almost the entire length of the beach before jumping in to swim, as the y distance is approximately 1/10 the distance x. This makes sense considering the running rate is roughly seven times faster than the swimming rate. This equation allows us to determine the point at which Lassie should enter the water, given the vertical distance out into the water (distance x) and assuming Lassie s running and swimming rates are consistent with those used in our analysis. 4

5 Part III Pennington wanted to test the relationship found in Part II to determine if Elvis knew calculus. Quoting from Pennington (2003), I took Elvis to Lake Michigan on a calm day when the waves were small. I fixed a measuring tape about 15 meters down the beach at C from where Elvis and I stood at A as I threw the ball. After throwing it, I raced after Elvis, plunging a screwdriver into the sand at the place where he entered the water at D. Then I quickly grabbed the free end of the tape measure and raced him to the ball. I was then able to get both the distance from the ball to the shore, x, and the distance y. If my throw did not land close to the line perpendicular to the shoreline and passing through C, I did not take measurements. I also omitted the couple of times when Elvis, in his haste and excitement, jumped immediately into the water and swam the entire distance. I figured that even an A student can have a bad day. WE spent three hours getting 35 pieces of data. We stopped only when the waves grew. Elvis had no interest in stopping or slowing down. The following data were collected. x y x y x y x y x y Using these data and your result from Part II, provide an argument regarding Elvis s innate ability to do calculus. To continue our analysis, we used the data you sent from the town mathematician Pennington s Lake Michigan experiment. We plotted the data Pennington collected on the same graph as the equation we found modeling the optimal relationship between distances x and y. This graph is seen below. 5

6 The red line on the graph shows the relationship between x and y that optimizes time. The blue data points correspond to the data Pennington collected. As you can see from the graph, the blue data points do appear to fall near the optimal y = x line. This is especially true at the lower values of x, when x is less than 10 meters. However, as x gets larger, notably above 15 meters, the blue data points do not fall near the optimal line. In fact, the y values are much larger than they should be according to the optimized model, meaning that the dog entered the water sooner than was necessary. From the models we created and the graph seen above, we can conclude that dogs like Lassie and Elvis are able to optimize the time it takes to rescue a drowning person or retrieve a floating ball from the lake. This is especially true when the object is less than 10 meters vertically from the shore. The dog s ability to optimize time does decrease as the distance to the object increases from the shore. However, we should consider the fact that the dogs are not using any type of measurement device, but are simply eyeballing how far to run down the beach. It is true, for dogs and humans alike, that it is easier to gauge smaller distances than larger ones. Therefore, we feel that the stray data points do not indicate that the dogs are unable to optimize their time, but are more a reflection of the dog s ability to gauge long distances correctly. Whether the dogs are minimizing their time by doing calculus, or have learned by experience how far to run down the beach before jumping in, that is not something we can determine. We can say though, from our analysis, that dogs are in fact able to optimize the time it takes to retrieve an object from the water. 6

7 Part IV There may be some limitations to this experiment. Conclude your report by commenting on any limitations or assumptions used in this project. After performing the analysis of the data you sent, we have determined that there are some limitations to this experiment. We were limited by the quality of the data sent from Pennington. As he was performing the experiment alone, trying to run after the dog to determine the dog s point of entry into the water, as well as racing the dog to the ball to obtain a measurement for distance x, we feel that his measurements are not as accurate as they could be. In order to improve the quality of the data, Pennington should have recruited some friends to help with the measurements. Had Pennington spaced assistants every 2-5 meters apart along the beach, the assistants could then have more accurately determined the point at which the dog entered the water. Additionally, the assistant at the end of the beach could have begun swimming out the ball immediately to obtain a more accurate measurement of the vertical distance x out to the ball. The use of assistants would greatly improve the quality of the data collected. Our analysis is also limited by the assumption that the dogs running and swimming rates remained constant throughout the course of the experiment. While this may be true, especially over smaller distances, it is likely that the running and swimming rates slowed over time. A more thorough analysis would take into account that the running and swimming rates may be changing over time. To improve this part of the experiment, Pennington could have equipped his assistants with stopwatches to record the dog s running and swimming times for each trial. This would give enough data to determine the running and swimming rates for each trial, rather than assuming these rates were constant throughout the entire length of the experiment. Despite these limitations, we still feel that this analysis gives us a good picture of a dog s ability to minimize time when retrieving an object from the water. We have always known that dogs are man s best friend. Now we can also say that dog s are math s best friend, too! However, we do not recommend that you start letting your dog do your homework. They have terrible penmanship! Note: This projected is adapted from an article written by Timothy J. Pennings titled Do Dogs Know Calculus?, published in The College Mathematics Journal, Vol. 34, No. 3, May 2003 and published by the Mathematical Association of America. Here is Elvis 7

8 Scoring Sheet The Team Project is a group activity in which the students are presented a series of mathematical problems relating to a specific theme. The team members are to solve the problems and write a narrative about the theme which answers all the mathematical questions posed. Teams are graded on accuracy of mathematical content, clarity of explanations, and creativity in their narrative. A holistic scoring approach should be used to judge the team project. For each project, assign a score to each of the major areas: School Name: - Accuracy of mathematical content: Comments: Clarity of Explanations: Comments: Creativity in Narrative: Comments: Overall Presentation: Comments: 8