Modeling Exponential Growth and Decay

Transcription

1 Lesson 7-4 Lesson 7-4 Modeling Eponential Growth and Deca Vocabular eponential regression BIG IDEA Situations of eponential growth and deca can be modeled b equations of the form = bg. As ou saw with the population of California in Lesson 7-2, sometimes a scatterplot shows a data trend that can be approimated b an eponential equation. Similar to linear regression, our calculator can use a method called eponential regression to determine an equation of the form = b g to model a set of ordered pairs. Modeling Eponential Deca In the Activit below, eponential regression models how high a ball bounces. Mental Math Each product is an integer. Write the integer. a b c d Activit Each group of or 6 students needs at least 3 different tpes of balls (kickball, softball, and so on), a ruler, markers or chalk, and large paper with at least 2 parallel lines that are 3 inches apart. Step 1 Tape the paper to a wall or door so the horizontal lines can be used to measure height above the fl oor. To make measuring easier, number ever fourth line (12 in., 24 in., 36 in., and so on). Step 2 One student will drop each ball from the highest horizontal line and the other students will act as spotters to see how high the ball bounces. The 1st spotter will mark the height to which the ball rebounds after the 1st bounce. The 2nd spotter will mark the rebound height after the 2nd bounce, and so on, until the ball is too low to mark. Step 3 Make a table similar to the one at the right and record the rebound heights after each bounce. Step 4 For each ball, enter the data in a list and create a scatterplot on our calculator. Bounce Ball Height (in.) (drop height)? 1? 2? 3??? Step a. Use the linear regression capabilit of our calculator to fi nd the line of best fi t for the data. Graph this line on the same screen as our scatterplot. Sketch a cop of the graph. (continued on net page) Modeling Eponential Growth and Deca 419

2 Chapter 7 b. Find the deviation between the actual and predicted height of the ball after the 3rd bounce. c. Use our linear regression model to predict the height of the ball after the 8th bounce. Step 6a. Use the eponential regression capabilit of our calculator to fi nd an eponential curve to fi t the data. Graph this equation on the same screen as our scatterplot. Sketch a cop of the scatterplot and curve. b. Find the deviation between the actual and predicted height of the ball after the 3rd bounce. c. Use our eponential regression model to predict the height of the ball after the 8th bounce. Step 7 Which seems to be the better model of the data the linear equation or the eponential equation? Eplain how ou made our decision. Step 8 Repeat Step 6 to fi nd eponential regression equations that fi t the bounces of the other balls. Step 9 Write a paragraph comparing the bounciness of the balls ou tested. QY Modeling Eponential Growth Advances in technolog change rapidl. Some people sa that if ou purchase a computer toda it will be out of date b tomorrow. When computers were first introduced to the public, the ran much more slowl. As computers have advanced over the ears, the speed has increased greatl. On the net page is an eample of data that a person collected to show the advancement in computer technolog. The processing speed of a computer is measured in megahertz (MHz). GUIDED Eample The table and graph on the net page show the average speed of a computer and the ear it was made. a. Write an equation to model the data. b. Find the deviation between the actual speed for the ear 2 and the predicted speed. c. Use the model to predict the processing speed of a computer made in 22. QY A student dropped a ball from a height of.912 meter and used a motion detector to get the data below. Rebound Bounce Height (m) a. Write an eponential equation to fi t the data. b. After the 8th bounce, how high will the ball rebound? 42 Using Algebra to Describe Patterns of Change

3 Lesson 7-4 Year Years since 1976 Speed (MHz) Speed (MHz) Year Source: Microprocessor Quick Reference Guide Solutions a. First enter the data into our calculator lists. Instead of letting ears be the -values, let = the ears since So for 1976 itself, = and for 1978, = 2. Net, use eponential regression on our calculator to fi nd an eponential equation to fi t the data. For = b g, the calculator gives b and g (Your calculator ma call this equation = ab.) The eponential equation that best fits the data is =?. b. For 2, = 24 and the actual speed was 42 MHz. Substitute 24 into the equation to fi nd the predicted value. The predicted speed is =? MHz. The deviation is 42 -? =?. The actual processing speed in 2 was? more than the predicted speed. c. The ear 22 is , or 44 ears after 1976, so substitute 44 for in our eponential equation. =? (?? ), so the predicted processor speed for the ear 22 is? MHz. Questions COVERING THE IDEAS 1. Suppose a ball is dropped and it rebounds to a height of feet after bouncing times, where = 6(.). Use the equation to a. give the height from which the ball was dropped, and b. give the percent the ball rebounds in relation to its previous height. On April 2, 1961, the patent offi ce awarded the fi rst patent for an integrated circuit to Robert Noce while Jack Kilb s application was still being analzed. Toda, both men are acknowledged as having independentl conceived of the idea. Source: PBS Modeling Eponential Growth and Deca 421

4 Chapter 7 In 2 4, use the graph to answer the questions. The percent written above each bar represents the percent of the previous height to which each tpe of ball will rebound. 2. Which ball s rebound height could be modeled b the equation = 1(.49)? 3. If a basketball is dropped from a height of 1 feet above the ground, how high will it rebound after the 1st bounce? After the th bounce? 4. Find and compare the rebound percentages in the Activit on page 419 to those in the graph. Are the similar or different?. A computer s memor is measured in terms of megabtes (MB). The table at the right shows how much memor an average computer had, based on the number of ears it was made after Use eponential regression to predict the amount of memor for a computer made in For each scatterplot, tell whether ou would epect eponential regression to produce a good model for the data. Eplain our reasoning. a. b. Percent (%) % Source: Eploratorium 32% 36% Table Tennis Ball Baseball Golf Ball 4% Soccer Ball 49% Tennis Ball Tpe of Ball Years After % Basketball 81% Rubber Ball Memor (MB) % Steel Ball on Steel Plate c. d. 422 Using Algebra to Describe Patterns of Change

5 Lesson 7-4 APPLYING THE MATHEMATICS Matching In 7 9, the graphs relate the bounce height of a ball to the number of times that it has bounced. Match a graph to the equation. a. =.1(.9) b. = 8.7(.9) c. = 8.7(.42) Rebound Height (ft) Bounce Number Rebound Height (ft) Bounce Number Rebound Height (ft) Bounce Number 1. The table at the right shows the number of weeks a movie had plaed in theaters, how it ranked, and how much mone it grossed each weekend. (Note that = is the weekend the movie opened.) a. Create a scatterplot with = gross sales after weeks in theaters. Wh is the eponential model a better model for these data than a linear model? b. Use eponential regression to find an equation to fit the data. c. What gross sales are predicted for the weekend of the 2th week? 11. Ldia and Raul started with 2 pennies in a cup, shook them out onto the table, and added a penn for each coin that showed a head. The continued to repeat this process and their data are recorded in the table at the right. a. Create a scatterplot of their data. b. Use eponential regression to derive an equation relating the trial number to the number of pennies the will have on the table. Trial Number Number of Pennies Weeks in Theaters Rank Weekend Gross ($) 1 114,844, ,417, ,36, ,8, ,317,411 1,311, ,1, ,, ,13, ,24, , ,186 Modeling Eponential Growth and Deca 423

6 Chapter 7 For 12 and 13, create a real-world problem that could be modeled b the given equation. 12. = 72(1.8) 13. = 14(.6) REVIEW 14. The population of a cit is 1,2,. Write an epression for the population ears from now under each assumption. (Lessons 7-3, 7-2, 6-1) a. The population grows 2.% per ear. b. The population decreases 3% per ear. c. The population decreases b 1, people per ear. 1. Graph = 12 ( 2 ) for integer values of from to. (Lesson 7-3) 16. Rewrite 3 4 using the Repeated Multiplication Propert of Powers. (Lesson 7-1) 17. An art store bus a package of 4 bristle paintbrushes for $8. and a package of 3 sable paintbrushes for $1. If the plan to sell an art kit with 4 bristle paintbrushes and 3 sable paintbrushes, how much should the charge for the kit to break even on their costs? (Lesson -3) 18. Skill Sequence Divide and simplif each epression. (Lesson -2) a. 4 b. 4 c Recall that if an item is discounted %, ou pa (1 - )% of the original price. Calculate in our head the amount ou pa for a camera that originall cost $3 and is discounted each indicated amount. (Lesson 4-1) a. 1% b. 2% c % 2. Evaluate (3a) 3 (4b) 2 when a = 2 and b = 6. (Lesson 1-1) EXPLORATION 21. In the ball-drop activit on pages and Questions 2 4 on page 422, ou eplored the rebound height of a ball as a percent of its previous height. Different tpes of balls have different percents. Does the height from which the ball is dropped affect the percent a ball will rebound? Eplain our answer. 22. Do the activit described in Question 11 on page 423. How close is our eponential model to the one in that question? a oung artist at work QY ANSWERS a. =.917(.816) b. approimatel.18 m 424 Using Algebra to Describe Patterns of Change