Roger Ranger and Leo Lion

Transcription

1 Concepts Slope and point-slope form of a line Distance between two points D = r*t Parametric equations Graphical interpretation Roger Ranger and Leo Lion Materials Student activity sheet Roger Ranger and Leo Lion TI InterActive! Introduction This activity engages students in using TI InterActive! to create a mathematical model that tracks the movement of a lion and a wildlife preserve ranger within a jungle and wildlife preserve. Students use linear and parametric equations to investigate whether or not Leo Lion will have Roger Ranger for dinner. The scenario begins with a jungle and wildlife preserve that extends 80 miles north and 120 miles east of a ranger station. Roger Ranger leaves from a point 100 miles east of the station along the southern boundary to survey the area. He travels 0.6 miles north and 0.5 miles west every minute. Leo Lion leaves the west edge of the preserve 51 miles north of the station at the same time the ranger leaves the station. Every minute Leo Lion moves 0.1 miles north and 0.3 miles east. Students determine whether Leo Lion and Roger Ranger collide and then model the movements of Leo Lion and Roger Ranger with parametric equations. PTE: Algebra Page 1

2 Student Activity Sheet Adapted from The Ranger and the Lion written by Sam Gough A jungle and wildlife preserve extends 80 miles north and 120 miles east of a ranger station. Roger Ranger leaves from a point 100 miles east of the station along the southern boundary to survey the area. He travels 0.6 miles north and 0.5 miles west every minute. Leo Lion leaves the west edge of the preserve 51 miles north of the station at the same time the ranger leaves the station. Every minute Leo Lion moves 0.1 miles north and 0.3 miles east. Use TI InterActive! to determine whether or not Leo Lion and Roger Ranger collide. Part I 1. Write a linear equation for the path traveled by the ranger and the lion by clicking on the Math Box and typing the following command for each. 2. Graph the paths by clicking on the graph icon. You should see the following two screens: Enter your first equation in the blank to the right of y1(x):= and then click in the white box at the beginning of the row containing y1(x):=. Repeat the process for entering the second equation in y2(x). Click on the colored squares or the line style blocks to make changes in color or line style. To adjust the window on the graph, click on the brown Format button on the Graph window and enter appropriate values. To label and scale the axes and title the graph, click on the Format button, select the Labels tab, and enter names for axes and graph title. Select the Windows tab and enter the scale factor for the axes. 3. Use the Trace feature on the Graph window to simulate the ranger s walk and the lion s walk and determine the point where their paths cross. When the large is located at the point of intersection, click on the Trace button and then click on the Label button. PTE: Algebra Page 2

3 4. How far is it from where Roger Ranger starts to where the paths cross? Write an equation to represent this distance. Hint: Click on the Math Box and type dr:= followed by the formula representing the distance Roger Ranger walks from when he begins to when their paths cross. 5. How far is it from where Leo Lion starts to where the paths cross? Write an equation to represent this distance. Hint: Click on the Math Box and type dl:= followed by the formula representing the distance Leo Lion walks from when he begins to when their paths cross. 6. At what rate is Leo Lion traveling per minute? Hint: In a Math Box use rl:= to define the rate at which Leo Lion walks. 7. At what rate is Roger Ranger traveling per minute? Hint: In a Math Box use rr:= to define the rate at which Roger Ranger walks. 8. How long does it take for Leo Lion and Roger Ranger to reach the point at which their paths cross? Hint: In a Math Box use tr:= to define the travel time of Roger Ranger and tl:= to define the travel time of Leo Lion. Part II 1. Based on the information given in the problem s introduction, complete the chart indicating the coordinates of Roger Ranger and Leo Lion relative to the ranger station. 2. The parametric equations that model the motion of Roger Ranger and Leo Lion are found in the last row of the completed table. Graph the paths followed by Roger Ranger and Leo Lion by selecting the parametric graphing mode (Edit Mode Settings or use the Mode icon) and then selecting the Graph icon. Enter the parametric equations for Roger Ranger East in x1(t):=, Roger Ranger North in y1(t):=, Leo Lion East in x2(t):=, and Leo Lion North in y2(t):=. To adjust the window on the graph, click on the brown Format button on the Graph window and enter appropriate values. To label and scale the axes and title the graph, click on the Format button, select the Labels tab, enter names for the axes and graph title, select the Windows tab, and enter the scale factor for each axis. PTE: Algebra Page 3

4 3. Use the Trace feature on the Graph window to simulate the ranger s and the lion s walks and determine the point where their paths cross. When the large is located at the point of intersection, click on the Trace button and then click on the Label button. 4. Compute the time it takes Roger Ranger to reach that point using a solve command with the East equation. Compute the time it takes Roger Ranger to reach that point using a solve command with the North Equation. How do the results compare? 5. Compute the time it takes Leo Lion to reach that point using a solve command with the East equation. Compute the time it takes Leo Lion to reach that point using a solve command with the North Equation. How do the results compare? 6. How do the times agree with the answers to Part I, question 8? Do Roger Ranger and Leo Lion collide? 7. If they don t collide, how close do they get to each other? Find the distance between Roger Ranger and Leo Lion at the following times. Copy the information from the table in Part II, question The distance between Ranger Roger and Leo Lion found as a function of time can be graphed parametrically. Enter t as x3(t) and the distance equation as y3(t). Trace to the point closest to the East-axis. Find the time to the nearest 1-minute interval and the distance that Roger Ranger and Leo Lion are the closest. PTE: Algebra Page 4

5 Teacher Notes Introduction This activity engages students in using TI InterActive! tocreate a mathematical model that tracks the movement of a lion and a wildlife preserve ranger within a jungle and wildlife preserve. Students use linear and parametric equations to investigate whether or not Leo Lion will have Roger Ranger for dinner. The scenario begins with a jungle and wildlife preserve that extends 80 miles north and 120 miles east of a ranger station. Roger Ranger leaves from a point 100 miles east of the station along the southern boundary to survey the area. He travels 0.6 miles north and 0.5 miles west every minute. Leo Lion leaves the west edge of the preserve 51 miles north of the station at the same time the ranger leaves the station. Every minute Leo Lion moves 0.1 miles north and 0.3 miles east. Students determine whether Leo Lion and Roger Ranger collide and then model the movements of Leo Lion and Roger Ranger with parametric equations. Instructions 1. Guide students in creating a TI InterActive! document to model the movement of Leo Lion and Roger Ranger. 2. Use a computer projection device to facilitate student progress through the activity. Answer Key Part I 1. Write a linear equation for the path traveled by the ranger and the lion. Roger Ranger: Leo Lion: 2. Graph the paths by clicking on the graph icon. PTE: Algebra Page 5

6 3. Use the Trace feature on the Graph window to simulate the ranger s walk and the lion s walk and determine the point where their paths cross. East 45 miles; North 66 miles 4. How far is it from where Roger Ranger starts to where the paths cross? Write an equation to represent this distance. Use the distance formula: 5. How far is it from where Leo Lion starts to where the paths cross? Write an equation to represent this distance. In miles: 6. At what rate is Leo Lion traveling per minute? The lion is walking north at a rate of 0.1 miles per minute and at a rate of 0.3 miles east per minute; the question is asking how fast is it moving along a diagonal line. Use the Pythagorean Theorem on the right triangle. In miles per minute: 7. At what rate is Roger Ranger traveling per minute? The ranger is walking north at a rate of 0.6 miles per minute and east at a rate of 0.5 miles per minute; the question is asking how fast he is moving along a diagonal line. Use the Pythagorean Theorem on the right triangle. In miles per minute: 8. How long does it take for Leo Lion and Roger Ranger to reach the point at which their paths cross? Ranger time in minutes: Lion time in minutes: PTE: Algebra Page 6

7 Part II 1. Based on the information given in the problem s introduction, complete the chart indicating the coordinates of Roger Ranger and Leo Lion relative to the ranger station. 3. The parametric equations that model the motion of Roger Ranger and Leo Lion are found in the last row of the completed table. Enter the parametric equations for Roger Ranger East in x1(t):=, Roger Ranger North in y1(t):=, Leo Lion East in x2(t):=, and Leo Lion North in y2(t):=. The parametric equations are: Ranger East North Lion East North x1(t):=100-.5t y1(t):=.6t x2(t):=.3t y2(t):=51+.1t 4. Use the Trace feature on the Graph window to simulate the ranger s and the lion s walks and determine the point where their paths cross. PTE: Algebra Page 7

8 The point of intersection (East, North) is (45, 66). 5. Compute the time it takes Roger Ranger to reach that point using a solve command with the East equation. Compute the time it takes Roger Ranger to reach that point using a solve command with the North Equation. How do the results compare? Roger Ranger's time of arrival in minutes using East equation: Roger Ranger's time of arrival in minutes using North equation: 6. Compute the time it takes Leo Lion to reach that point using a solve command with the East equation. Compute the time it takes Leo Lion to reach that point using a solve command with the North Equation. How do the results compare? Leo Lion s time of arrival in minutes using East equation: Leo Lion s time of arrival in minutes using North equation: 7. How do the times agree with the answers to Part I, question 8? Do Roger Ranger and Leo Lion collide? Using the distance formula that distance = rate * time: No, they do not collide. 8. If they don t collide, how close do they get to each other? Find the distance between Roger Ranger and Leo Lion at the following times. Copy the information from the table in Part II, question 1. (NOTE: The input variable must be changed from t to m (or another letter) in order to get the expanded equation. Since t has already been assigned a value, a computed value for d is given.) PTE: Algebra Page 8

9 8. The distance between Ranger Roger and Leo Lion found as a function of time can be graphed parametrically. Enter t as x3(t) and the distance equation as y3(t). Trace to the point closest to the East-axis. Find the time to the nearest 1-minute interval and the distance that Roger Ranger and Leo Lion are the closest. The closest they come is 9.75 miles. (minutes, miles) = ( , 9.749) PTE: Algebra Page 9