LAB 2: LINEAR MODELING

Transcription

1 LAB 2: LINEAR MODELING Objectives: 1. Create linear models from real data. 2. Use interpolation and extrapolation; analyze and evaluate results. 3. Read, analyze and interpret graphs. 4. Find average rates of change. Reference Topic: Linear Functions Linear Models Discussion: In this lab, you will create linear models (equations) based on real data. Then you will use the models to make predictions. In addition, you will review and apply the concept of average rate of change. Predicting Eruptions of Old Faithful Yellowstone National Park contains over half of the world s geysers. A geyser is a hot spring with intermittent bursts of water or steam. Old Faithful is the most famous geyser at Yellowstone. It erupts for between and 5 minutes at heights reaching more than 180 feet, and visitors need wait only 1 2 to 2 hours to see an eruption. Through extensive observation and data gathering, it was discovered that the interval between eruptions of Old Faithful depends on the duration or length of the eruption. Old Faithful s eruption intervals have been observed more closely than any other geyser s at Yellowstone....Since 1920 the National Park Service has stationed park rangers at Old Faithful. One of their duties is to determine the time of each daylight eruption in order to predict for visitors the approximate time of the geyser s next performance. Each year, 2000 to 3500 eruptions are visually timed and recorded. 1 How do Yellowstone rangers predict eruptions of Old Faithful? They observe and time the length of an eruption, and predict the time of the next eruption using the chart below (Source: Yellowstone National Park, 1993). Length of Eruption (in minutes) Time Until Next Eruption (in minutes) Marler, G.D. (1988), The Story of Old Faithful, Yellowstone Association, Yellowstone National Park, page 7.

2 a. Enter the table on DERIVE or the TI-82. Let x represent the length of the eruption. Let y represent the number of minutes until the next eruption. DERIVE TI-82 Reminder: Type D for Declare, then M for Matrix. Press 8 TAB 2 ENTER for 8 rows and 2 columns. Then enter the data by row, pressing ENTER between each entry. That is, type 1.5 ENTER 51 ENTER etc. When you have entered all the given data, you should see the complete table in the Algebra window. OR Reminder: Press STAT 1 to edit your lists. Enter the x-data in L1 and the corresponding y-data in L2. Enter the data by columns, pressing ENTER between each entry. That is, type 1.5 ENTER 2 ENTER etc. Use the arrow keys to move between lists. b. Plot the data using DERIVE or the TI-82. Be sure to use an appropriate scale. TI-82 Reminder: Press 2nd Y= for STATPLOT, then 1 for Plot1. Press ENTER to turn Plot1 on, then use the arrow and ENTER keys to highlight the window as follows: c. Notice that the data seem to fall into a linear pattern. Make a copy of the graph of the points for each member of your lab team. d. Working alone, each member of your lab team should draw a line that he or she thinks fits the data well and find the equation of his or her line. e. Each member should plot his or her equation (linear model) using DERIVE or the TI-82, on the same screen as the original data. If you detect an error, (if the line graphed is not the line you expected to see), the team member should recompute the equation and regraph it. Your team should end up with two different equations that closely fit the data points. If you have more than two members on your team, pick the two models that appear to best match the data. Turn in a graph for each of the two models that includes the data and the line representing the model. Be sure to label your axes, indicate the scale, and title the graphs. 2-2

3 f. What are your two models (equations)? Model 1: y = Model 2: y = g. What is the domain for your models? In other words, what interval of input values makes sense in the context of this problem? h. Complete the following chart using the two different models you have created. DERIVE TI-82 Reminder: Use the vector statement. For example, to make a table of values for model y = 2x + 3, with x going from 1.5 to 5 at.5 unit intervals, the vector statement would be vector([x, 2x+3], x, 1.5, 5,.5) OR Reminder: Press 2nd WINDOW for TblSet. Set TblMin=1.5 and DTbl=.5. Be sure that Indpnt and Depend are set to Auto. Then press 2nd GRAPH for TABLE. Predicting Eruptions of Old Faithful (Times are in Minutes) LENGTH OF ERUPTION TIME UNTIL NEXT ERUPTION Given Model 1 Prediction Model 2 Prediction 2-3

4 i. Using only the information in the chart in part (h), determine which one of your models, Model 1 or Model 2, gives the best predictions ( best fit ). Show how you reached your conclusion. Be as specific as possible. j. You can use your model to interpolate y values for x values within the given domain of the data. Using the model that gave the best fit, find the time until the next eruption if the eruption lasts A minutes B. 3 minutes and 42 seconds +NOTE: Remember that 3 min. 42 sec. is not the same as 3.42 min. 3 min. 42 sec.=3 42 =3.7 min. 60 C. 4 minutes and 20 seconds k. Does the time between eruptions increase or decrease as the length of eruptions increases? At what rate? 2-4

5 Winning the Indianapolis 500 The Indianapolis 500 (Indy 500) is one of the most famous auto races in the world. It has taken place almost every year since As the name implies, the Indy 500 is a 500 mile race covering 200 laps of the 2.5 mile Indianapolis Motor Speedway. The first winner took over 6 hours to complete the race. Since that first Indy 500, advances in technology have produced lower, more aerodynamic, more powerful, and thus much faster Indy race cars. By 1990, the winner completed the Indy 500 in less than 3 hours, averaging almost 190 mph. The following table gives the times in which the winning race car finished the Indy 500 for the given years. (Source: 1992 Information Please Almanac) Year Winner Car Time (hr:min:sec) 1920 Gaston Chevrolet Monroe 5:38: Billy Arnold Miller-Hartz Special 4:58: Wilbur Shaw Boyle Special 4:22: Johnnie Parsons Wynn s Friction Proof Special Race ended at 345 miles because of rain 1960 Jim Rathmann Ken-Paul Special 3:36: Al Unser 1980 Johnny Rutherford 1990 Arie Luyendyk Johnny Lightning P.J. Colt-Ford Penzoil Chaparral-Cosworth Domino s Pizza Lola-Cosworth 3:12: :29: :41: a. Plot the data using DERIVE or the TI-82. Let the independent variable represent the year with x = 0 representing 1900, x = 20 representing 1920, etc. and let the dependent variable (y) represent the winning time in hours. b. Copy the resulting graph and draw the line that you think best fits the data. Find the equation of your best fit line. Graph your equation using DERIVE or the TI-82 to make sure you get the line you expect. Correct your equation if necessary. Write your equation in the space below. c. Turn in a copy of the graph of the data along with your best fit line. Be sure to label your axes, indicate the scale, and title the graph. 2-5

6 d. You can use your model to extrapolate y values for x values outside the given domain for the data and thus make predictions about the winning race times for future years. Using your model, predict the winning time for the Indy 500 in the given years. Write your answer in hours and minutes (3 hours 15 minutes, rather than 3.25 hours). A. in 1999 B. in 2010 e. Is the winning time increasing or decreasing? Use your model to determine the average rate at which the winning time is changing. f. According to your model, what is the first year in which the winner will complete the Indy 500 in 2 hours or less? g. Find the x-intercept for the line you created to model this data. Interpret the meaning of the x-intercept for this data. Notice that the x- intercept gives the year in which the race will be finished in no time. In what year does your model predict that the race will be completed in no time? h. Based on parts (c) - (g), do you think a linear model is a good predictive model for winning race times for any year for the Indianapolis 500? For what domain (years) could you use this model to give you a good estimate of winning race times? Give reasons for your answers. - i. On a separate sheet of paper, write 1-2 paragraphs explaining to an algebra student who has not done this lab how to model a set of real data using a linear model. Include appropriate uses and limitations of such a model. 2-6