1.2. Modeling Growth and Decay. Launch LESSON 1.2. Vocabulary decay growth principal simple interest compound interest. Materials

Transcription

1 LESSON LESSON.2 Modelig Growth ad Decay.2 Each sequece you geerated i the previous lesso was either a arithmetic sequece with a recursive rule i the form u = u + d or a geometric sequece with a recursive rule i the form u = r u. You compared cosecutive terms to decide whether the sequece had a commo differece or a commo ratio. This lesso gives more experiece with geometric sequeces, both decreasig ad icreasig. I most cases you have used u as the startig term of each sequece. I some situatios, it is more meaigful to treat the startig term as a zero term, or u. The zero term represets the startig value before ay chage occurs. You ca decide whether it would be better to begi at u or u. COMMON CORE STATE STANDARDS APPLIED DEVELOPED F.IF.3 F.BF.2 F.LE.c F.LE.2 INTRODUCED EXAMPLE A Objectives Discover applicatios ivolvig geometric sequeces Use geometric sequeces to model growth ad decay situatios Uderstad the physical limitatios of models The Kelley Blue Book, first compiled i 926 by Les Kelley, aually publishes stadard values of every vehicle o the market. May people who wat to kow the value of a automobile will ask what its Blue Book value is. The Kelley Blue Book calculates the value of a car by accoutig for its make, model, year, mileage, locatio, ad coditio. Vocabulary decay growth pricipal simple iterest compoud iterest Materials u = 23, 999 u =.8 u where 38 CHAPTER After 6 years, the car is worth $6, Liear Modelig Startig value. 4 is.8. Use this rule to fid the 6th term. Lauch i. The temperature of hot chocolate ii. Food left i a locker iii. Amout of moey i a savigs accout iv. Value of a car v. Height of a boucig ball vi. Populatio of a city 4 of what it was worth the previous year. Therefore Each year, the car will be worth the sequece has a commo ratio, which makes it a geometric sequece. It is coveiet to start with u = 23,999 to represet the =.8*B7 fx value of the car whe it was ew so that u will represet the value after year, ad so o. The A B recursive formula that geerates the sequece of u aual values is Solutio balls (racquetballs, basketballs, or hadballs work well) video camera paper ad colored markers motio sesors or meter sticks, optioal Calculator Notes: Lookig for the Reboud Usig the EasyData App; Eterig Data; Plottig Data; Tracig Data Plots; Sharig Data Which of the followig ca be modeled as growth ad which ca be modeled as decay? Explai. A automobile depreciates, or loses value, as it gets older. Suppose that a particular automobile loses oe-fifth of its value each year. Write a recursive formula to fid the value of this car whe it is 6 years old, if it cost $23,999 whe it was ew. C h a p t e r Liear Modelig I situatios like the problem i Example A, it s easier to write a recursive formula tha a equatio usig x ad y. ELL Support Advaced You might start by askig whether the price of a car should go up, dow, or stay the same as it gets older ad why. What if the car is a classic sports car i mit coditio? Lik this to growth ad decay. Check readig ad verbal comprehesio of terms like growth, decay, reboud, etc. Sped time o the Lauch, discussig whe each situatio might be a example of growth ad whe it might be a example of decay. Lik the specific otatio of recursive rules (u otatio) with the verbal descriptio of a sequece by usig startig value ad rule. Use the Whole Class versio of the Ivestigatio. Focus o clear uderstadig of the specific otatio of recursive rules. As a extesio or challege, you might explore fidig a rule based o u ad u2 or u ad u4. Use the Oe Step versio of the Ivestigatio. DAA3_SE_CH_Priter_FINAL.idb 38 7//6 :4 AM

2 Step Height Bouce Reboud Number Height (m) Step 3 Step 3.9 Height Height Step Time YOU WILL NEED a ball a video camera paper to make ruler color markers Bouce Bouce Step u =. ad u =.64 u where ;.,.64,.4,.26,.7,. INVESTIGATION Lookig for the Reboud Whe you drop a ball, the reboud height becomes smaller after each bouce. I this ivestigatio you will write a recursive formula for the height of a real ball as it bouces. Step Step 2 Scroll through the video to record the iitial height ad subsequet heights whe the ball reaches the top of each bouce. You will ot be able to read the umbers o the ruler but should be able to use the colors to calculate the height. Trasfer the data to your calculator i the form ( x, y ), where x is the time sice the ball was dropped, ad y is the height of the ball. Trace the data graphed by your calculator to fid the startig height ad the reboud height after each bouce. Record your data i a table. Step 3 Graph a scatter plot of poits i the form ( bouce umber, reboud height ). Record the graphig widow you use. Step 4 Step Step 6 Compute the reboud ratio for cosecutive bouces. reboud height reboud ratio = sample aswer: previous reboud height.77,.7,.6, Decide o a sigle value that best represets the reboud ratio for your ball. Use this ratio to write a recursive formula that models your sequece of reboud height data, ad use it to geerate the first six terms. Compare your experimetal data to the terms geerated by your recursive formula. Aswers will vary. a. How close are they? Modifyig the Ivestigatio Whole Class Have two studets collect data i frot of the class. Complete Steps 3 with studet iput. Discuss Step 6. Shorteed Use the sample data. Have studets complete Steps 3. Discuss Step 6. Oe Step Give the istructios for the ivestigatio ad pose this problem: What is a recursive formula for the height of the ball at the top of each bouce? Procedure Note Collectig Data. Create a color ruler by markig each cetimeter with rotatig colors (Use at least three colors), clearly mark each cm. Attach the ruler to a wall. 2. Sit or keel with video camera at least 3 meters ( feet) from the wall. 3. Drop the ball as close to the wall as you ca. Record the iitial drop ad the first few bouces (approximately secods). b. Describe some of the factors that might affect this experimet. (For examples how might the formula chage if you used a differet kid of ball.) c. Accordig to the recursive formula does the ball ever stop boucig? d. Realistically, how may bouces do you thik there were before the ball stopped? Lesso.2 Modelig Growth ad Decay 39 After studets gather data, some will look for a additive formula (a arithmetic sequece) ad others for a multiplicative formula (a geometric sequece). Have studets discuss which models are best, based o differeces betwee predicted ad actual data. ASK Does your model have a limit o the umber of bouces before the ball stops? How realistic is that? Aswers could vary. The descriptio will dictate the aswer. For example, the populatio of a city could be either, depedig o whether people are movig ito or out of the city. Ivestigate Example A Cosider projectig the example from your ebook ad havig studets work i pairs. Have studets share their strategies alog with their solutios. Emphasize the use of correct mathematical termiology ad symbols. Whether studet resposes are correct or icorrect, ask other studets if they agree ad why. SMP, 3, 6 ASK Why ca we fid a 2% depreciatio by multiplyig by 4? If ecessary, suggest that they thik about specific umbers, such as $. Poit out that Example A seems to be claimig that 23,999 (.2)23,999 = 23,999(.8). ASK Is that reasoig justified? Explai. If eeded, suggest they factor the equatio. You might geeralize to u (.2) u = u (.2) = u (.8) as you review factorig. SMP 7 ASK Accordig to your rule, how low could the value of the car go? Realistically, what is the lowest value of the car? Guidig the Ivestigatio You ca use the Ivestigatio Worksheet Lookig for the Reboud with Sample Data if you do ot wish to coduct the ivestigatio as a activity. There is also a desmos simulatio of this ivestigatio i the ebook. If you have motio sesors, studets ca collect data with the motio sesor. A Ivestigatio Worksheet is available for use with motio sesors. Itroduce the ivestigatio by demostratig the bouce measuremet process. If you are usig small balls, balls without seams, such as racquetballs, will work best. ALERT If the camera is too close to the ball or the ball is too far from the wall you get a bad lie of sight ay time the ball is above or below the camera. LESSON.2 Modelig Growth ad Decay 39

3 Step You ca t really read measuremets marked o the wall but, if your backgroud is distict, you ca measure the peak heights by slowly scrollig through the video. Try to have a iitial height of about 2 meters. To miimize the parallax error from the camera agle you wat the camera at the height of the ball whe it is at the top of the bouce. As adjustig the camera height is difficult, you wat the camera to be far away from the ball ad the ball close to the ruler. You will have to traslate the colors ito umbers as you will ot be able to read the ruler i the video. Step 4 Let studets decide what amout of error i the data is acceptable. Have groups preset their work, choosig presetatios to iclude a variety of decimal places i the results, ad questio which umber of decimals is most appropriate. SMP 3,, 6 ASK Could you use the first or secod reboud height as u? [Yes, either; startig the sequece from the secod height would produce a idetical set of subsequet bouces.] Example B Cosider projectig the example from your ebook ad havig studets work i pairs. Have them share their solutios. Emphasize the use of correct mathematical termiology ad symbols. Whether studet resposes are correct or icorrect, ask other studets if they agree ad why. SMP, 3, 6 Summarize Have studets preset their solutios ad explai their work i the Examples ad the Ivestigatio. Ecourage a variety of approaches. Studets may wat to draw graphs or work with formulas. Durig the discussio, ASK What kid of sequeces did you see i Examples A ad B ad i the ivestigatio? [geometric sequeces] Emphasize the use of correct mathematical termiology. Whether studet resposes are correct or icorrect, ask other studets if they agree ad why. SMP, 3,, 6 EXAMPLE B Solutio 4 Chapter Liear Modelig You may fid it easier to thik of the commo ratio as the whole,, plus or mius a percet chage. I place of r you ca write ( + p) or ( p). The car example ivolved a 2% (oe-fifth) loss, so the commo ratio could be writte as (.2). Your boucig ball may have had a commo ratio of.7, which you ca write as (.2) or a 2% loss per bouce. I Example A, the value of the car decreased each year. Similarly, the reboud height of the ball decreased with each bouce. These ad other decreasig geometric sequeces are examples of decay. These examples of real world decay ca be modeled with a geometric recursive rule but every model has error or variatio i the applicatio of the model. The value of the car will ever be zero ad the ball does ot keep boucig forever, yet the mathematical model tells us that the value of the car will keep goig dow ad that the ball will still be boucig after dozes of bouces. So, we ca use models to uderstad ad predict behavior, but every value from the sequece is oly a estimate of the true value. The ext example is oe of growth, or a icreasig geometric sequece. Iterest is a charge that you pay to a leder for borrowig moey or that a bak pays you for lettig it ivest the moey you keep i your bak accout. Simple iterest is a percetage paid o the pricipal, or iitial balace, over a period of time. If you leave the iterest i the accout, the i the ext time period you will receive iterest o both the pricipal ad the iterest that were i your accout. This is called compoud iterest because you are receivig iterest o the iterest. Gloria deposits $2, ito a bak accout that pays 7% aual iterest compouded aually. This meas the bak pays her 7% of her accout balace as iterest at the ed of each year, ad she leaves the origial amout ad the iterest i the accout. Whe will the origial deposit double i value? The balace starts at $2, ad icreases by 7% each year. u = 2 u = u +.7 u where The recursive rule that represets 7% growth. u = ( +.7 ) u where Factor. Use techology, such as a spreadsheet or calculator, to compute year-ed balaces recursively. Term u is 429.7, so the ivestmet balace will more tha double i years. fx =.7*B2 ADVANCED You might discuss the differece betwee omial ad effective iterest rates. If a 6.% aual iterest rate is compouded mothly, 6.% is the omial iterest rate ad ( ) is the effective aual iterest rate. A 2 3 B u $2,. $2,4. $2,289.8 $2,4.9 fx =.7 * B2 A 9 2 B u $3, $3,934.3 $4,29.7 $4,4.38 Coceptual/Procedural Coceptual The real world examples ad Ivestigatio situatios help studets coceptualize icreasig ad decreasig geometric sequeces. Additioally, addig the compoet of lookig at the limits of a mathematical model compared to the physical world makes aalyzig ad evaluatig the model more coceptual. Procedural Studets practice the procedure of writig ad evaluatig geometric sequeces. 4 CHAPTER Liear Modelig

4 Accout balace ($) 6, 4, 2,.2 Exercises Practice Your Skills Time (yr). Fid the commo ratio for each sequece, ad idetify the sequece as growth or decay. Give the percet chage for each. a.,, 22, 337., 6.2,... a b , 29.37,.7, 4.7,.88,....; growth; % icrease.4; decay; 6% decrease c. 8., 82.4, 84.87, 87.42, 9.4,... d. 28., 9.36, 76., 6.97,....3; growth; 3% icrease.92; decay; 8% decrease 2. Write a recursive formula for each sequece i Exercise. Use u for the first term ad fid u. a 3. Write each sequece or formula as described. a. Write the first four terms of the sequece that begis with 2 ad has the commo ratio.. a 2, 2, 22, 23.2 b. Write the first four terms of the sequece that begis with ad decays % with each term. What is the commo ratio?, 42, 362., 37.62; commo ratio =.8 c. Write a recursive formula for the sequece that begis 2, 3, 48, 74.64,.... a = 2, a = a ( +.8) where Compoud iterest has may applicatios i everyday life. The iterest o both savigs ad loas is almost always compouded, ofte leadig to surprisig results. This graph ad spreadsheet show the accout balace i Example B. fx =B8*(+.7) A B u Leavig just $2, i the bak at a good iterest rate for years ca double your moey. I aother 6 years, the $2, will have tripled. Some baks will compoud the iterest mothly. You ca write the commo ratio as ( ) to represet oe-twelfth of the aual iterest, compoudig mothly. Whe you do this, represets moths istead of years. How would you chage the rule to show that the iterest is compouded 2 times per year? What would represet i this situatio? ( ); would represet the umber of weeks You will eed a graphig calculator for Exercise 8. Films quickly display a sequece of photographs, creatig a illusio of motio CRITICAL QUESTION How do the geometric sequeces i the Ivestigatio, Example A, Example B compare with oe aother? BIG IDEA The automobile depreciatio i Example A ad the boucig ball i the ivestigatio give a decreasig geometric sequece, described as decay. The bak accout i Example B shows a icreasig geometric sequece, described as growth. CRITICAL QUESTION How would you describe the differeces betwee the recursive formulas for geometric growth ad decay sequeces? BIG IDEA I growth, the multiplier is greater tha ; i decay, it s less tha. Formative Assessmet As studets work ad preset, assess their uderstadig of geometric sequeces ad commo ratios, as well as their ability to calculate ratios. Observe how studets relate situatios to otatio. Do they have a good uderstadig of the meaig of u cotext? Ca they represet a loss of 2%? Apply Extra Example. You buy a pair of limited editio shoes, the immediately sell them o a olie auctio site. The biddig starts at $ ad each bid pushes the price up by %. Make a table. If the th bidder purchases the shoes, how much does that perso pay? $3.79 Exercise ALERT Studets might offer the commo ratio as the percet chage. 2a. u = ad u =. u where ; u b. u = ad u =.4 u where ; u.77 2c. u = 8. ad u =.3 u where ; u 7.3 2d. u = 28. ad u =.92 u where ; u 9.3 Lesso.2 Modelig Growth ad Decay 4 2. Rewrite the expressio u + 2 u so that the variable appears oly oce. 3u Closig Questio Write a recursive formula for the height of a ball that is dropped from cm ad has a 6% reboud ratio. u = ; u =.6 u where Assigig Exercises Suggested:, 4 6 Additioal Practice: 2, 3, 7 2 LESSON.2 Modelig Growth ad Decay 4

5 Exercises 6, 7 These exercises are related to the ivestigatio. Studets who missed the ivestigatio may eed assistace visualizig the graph of the data. 6d. Yes, whe watchig a ball bouce from iches the ball stops movig before it has bouced betwee 2 to 3 times. Exercises 7, 8 If studets have difficulty uderstadig the recursive formulas because they are prited o oe lie, suggest that they write them out as they ve see them before. ELL Havig studets describe the real-world meaigs for these exercises will give them the chace to practice their vocabulary ad will also serve as a checkpoit for comprehesio. 7. is the iitial height, but the uits are ukow..2 is the percet loss, so the ball loses 2% of its height each reboud. Exercise 8 Because is related to the year 2, studets will probably coclude that the iterest rate is aual. Be ope to multiple iterpretatios:.2 could be of 3% 2 compouded mothly or 4 of % compouded quarterly. With these two optios would eed to chage to represet moths or quarters. Exercise 9 ALERT Discourage studets from aswerig with fractios of people. 4. Match each recursive rule to a graph. Explai your reasoig. a. u = ii. decay b. u = i. growth u = (.2) u where a u = ( +.2) u where c. u = iii. costat u = u where i. u ii. u iii Factor these expressios so that the variable appears oly oce. For example, x +. x factors ito x ( +.). a. x + Ax a b. A.8 A a c. x +.82 x d. 2u.8 u x ( + A ) (.8) A, or.82 A ( +.82) x, or.82 x (2.8) u, or. u Reaso ad Apply 6. Suppose the iitial height from which a rubber ball drops is i. The reboud heights to the earest ich are 8, 64,, 4,.... a. What is the reboud ratio for this ball?.8 b. What is the height of the teth reboud? i. c. After how may bouces will the ball reboud less tha i.? Less tha. i.? 2 bouces; 3 bouces d. Is there reaso to suspect that these last two estimates are ot correct? Explai. 7. Suppose the recursive formula u = ad u = (.2) u where models a boucig ball. Give real-world meaigs for the umbers ad Suppose the recursive formula u2 = 2, ad u = ( +.2) u where 26 describes a ivestmet made i the year 2. Give real-world meaigs for the umbers 2, ad.2, ad fid u29. a $2, was ivested at 2.% aual iterest i 2. u29 = $27, APPLICATION A compay with 2 employees is growig at a rate of 2% per year. It will eed to hire more employees to keep up with the growth, assumig its busiess keeps growig at the same rate. a. How may people should the compay pla to hire i each of the ext years? umber of ew hires for ext years: 2, 3, 3 (or 4), 4, ad b. How may employees will it have years from ow? about 3 employees. APPLICATION The table below shows ivestmet balaces over time. u 3 Elapsed time (yr) Balace ($) 2, 2,7 2,34.4 2, a. Write a recursive formula that geerates the balaces i the table. a u = 2; b. What is the aual iterest rate? 8.% u = ( +.8) u where c. How may years will it take before the origial deposit triples i value? 42 Chapter Liear Modelig 4 years ($6,266.8) 42 CHAPTER Liear Modelig

6 . APPLICATION Suppose you deposit $ ito a accout that ears 6% aual iterest. You do t withdraw or deposit ay additioal moey for 3 years. a. If the iterest is paid oce per year, what will the balace be after 3 years? a $9. b. If the iterest is paid every six moths, what will the balace be after 3 years? This is also referred to as 6% compouded semiaually. Divide the aual iterest rate by 2 to fid the semiaual iterest rate. $97.3 c. What will the balace be if you receive 6% compouded quarterly for 3 years? $97.8 d. What will the balace be if you receive 6% compouded mothly for 3 years? $ APPLICATION Suppose $ is deposited ito a accout that ears 6.% aual iterest ad o more deposits or withdrawals are made. a. If the iterest is compouded mothly, what is the mothly rate? % b. What is the balace after moth? $2.7 c. What is the balace after year? $33.49 d. What is the balace after 29 moths? $ Suppose Jill s biological family tree looks like the diagram at right. You ca model recursively the umber of people i each geeratio. a. Make a table showig the umber of Jill s acestors i each of the past five geeratios. Use u to represet Jill s geeratio. b. Look i your table at the sequece of the umber of acestors. Describe how to fid u if you kow u. Write a recursive formula. c. Fid the umber of the term of this sequece that is closest to billio. What is the real-world meaig of this aswer? Mom s mom Mom s dad Dad s mom Dad s dad d. If a ew geeratio is bor every 2 years, approximately whe did Jill have billio livig acestors i the same geeratio? 7 years ago Her mom Her dad e. Your aswer to 3c assumes there are o duplicates, Jill that is, o commo acestors o Jill s mom s ad Jill s dad s sides of the family. Look up Earth s populatio for the year you foud i 3d. Describe ay problems you otice with the assumptio of o commo acestors. The populatio of the plaet at the time was less tha billio. Jill must have some commo acestors. Exercise As eeded, remid studets that as the iterval betwee paymets (ad therefore the iterest rate) decreases, the umber of paymets icreases. Exercise 3 Ecourage studets to critique their aswer i 3c ad to compare it with their aswer i 3d. SUPPORT Studets may eed to be remided why u should be used to represet Jill s geeratio, especially because her parets are cosidered to be oe geeratio removed from Jill, u. Studets are sometimes ucertai whe to use u versus u as the startig value i a problem. Explai that the specific otatio depeds o the cotext of the problem ad the questio that is beig cosidered. ALERT This exercise ca geerate iterestig class discussios. However, studets may be sesitive about issues surroudig acestry, ad traditioal ad otraditioal families. Family trees are lists of family descedats ad are used i the practice of geealogy. People who research geealogy may wat to trace their family s medical history or atioal origi, discover importat dates, or simply ejoy it as a hobby. Author Alex Haley (92 992), hoored here i this statue i Aapolis, Marylad, told the powerful history of his family s prologed slavery ad decades of discrimiatio i his 976 geealogical book, Roots: The Saga of a America Family. The book, alog with the 977 televisio miiseries, ispired may people to trace their family lieage. 3b. Start with ad recursively multiply by 2; u = ad u = 2 u where. 3c. u3 ; 3 geeratios ago, Jill had billio livig acestors Lesso.2 Modelig Growth ad Decay 43 3a. Geeratios back Acestors i the geeratio u LESSON.2 Modelig Growth ad Decay 43

7 4. u = ad u =.88u where ; u 2 =.48, or 4.8%. It would take about 2, years to reduce to %. Cotext Guitar Feedback Some studets will recogize feedback as the loud whie a microphoe sometimes makes. Explai that this occurs whe the output of the speakers becomes the iput to the microphoe. The feedback is the result of the system beig uable to hadle this recursive process. 4. APPLICATION Carbo datig is used to fid the age of aciet remais of oce-livig thigs. Carbo-4 is foud aturally i all livig thigs, ad it decays slowly after death. About.4% of it decays i each -year period of time. Let %, or, be the begiig amout of carbo-4. At what poit will less tha % remai? Write the recursive formula you used.. APPLICATION Betwee 98 ad 2, the populatio of Grad Traverse Couty i Michiga grew from 4,899 to 86,999. a. Fid the actual icrease ad the percet icrease over the 3-year period. a 32, ad 8.47% b. If you average that total ad rate over 3 years, what is the Athropologist carefully chage per year? Fid the average rate of chage ad the average revealig huma remais percet rate of chage.,7 people ad.9% at a aciet burial site. u c. Sice the growth is ot liear, the actual chage each year is 98 = 4,899 ad u = ( +.9)u where 98 yields 97,99. differet. However, overall it averages out to the value you foud i b. What happes if you use the average percet rate you foud i b startig i 98 for 3 years? d. Try this agai with a rate of.% for 3 years. 87,89 people much closer e. Usig.%, fid the populatio estimates for 99 ad 2. u 99 = 64,27, u 2 = 74,673 f. Usig 98 ad the populatio estimates from e, what is the average rate of chage i people per year betwee 98 ad 99? 98 ad 2? 99 ad 2? 92; 989; 64 people per year g. Without computig, which do you thik is larger: ii. While the % rate is fixed the umber of i. The average rate of chage from 2 to 2, or people added icreases each year so the average from the secod te years will be higher ii. the average rate of chage betwee 99 ad 2? tha the average over the whole 2 years. Explai your thikig. 6. Taoufik looks at the secod problem of his wet homework that fell i a puddle. a. What is the commo ratio? How did you fid it? b. What are the missig terms? c. What is the aswer he eeds to fid? 6a. 3; 62 8 = 9, 32 = 9 6b. 2, 6,, 4,, 486, 48,, 322 6c. 8,98 Guitar feedback is a real-world example of recursio. Whe the amplifier is tured up loud eough, the soud is picked up by the guitar ad amplified agai ad agai, creatig a feedback loop. Jimi Hedrix (942 97), a pioeer i the use of feedback ad distortio i rock music, remais oe of the most legedary guitar players of the 96s. 44 Chapter Liear Modelig 44 CHAPTER Liear Modelig

8 Review 7. The populatio of the Uited States grew 9.7% from 2 to 2. The populatio reported i the 2 cesus was 38.7 millio. What populatio was reported i 2? Explai how you foud this umber millio; ( +.97) x = A elevator travels at a early costat speed from the groud level to a observatio deck at 6 m. This trip takes 4 s. The elevator s trip back dow is also at this same costat speed. a. What is the elevator s speed i meters per secod? 4 m/s b. How log does it take the elevator to reach the restaurats, located 4 m above groud level? a s c. Graph the height of the elevator as it moves from groud level to the observatio deck. d. Graph the height of the elevator as it moves from the restaurat level, at 4 m, to the observatio deck. e. Graph the height of the elevator as it moves from the deck to groud level. 8c. 8d. Height (m) Height (m) Time (s) 9. Cosider the sequece 8, 73, 66, 9,.... a. Write a recursive formula. Use u = 8. u = 8 ad u = u 7 b. What is u? where 2 u = 7 c. What is the first term with a egative value? u 27 = 2 2. Solve each equatio. a x = a x b..88x = x = 68. c. 8.7x 6 = x.83 The CN Tower i Toroto is oe of Caada s ladmark structures ad oe of the world s tallest buildigs. Built i 976, it has six glass-froted elevators that allow you to view the ladscape as you rise above it at mi/h. At 36 ft, you ca either brace agaist the wid o the outdoor observatio deck or test your erves by walkig across a 26 ft 2 glass floor with a view straight dow. 8e. Height (m) 6 4 Time (s) d x = 6 x = 8 2. For the equatio y = x, fid the value of y whe a. x = y = 47 b. x = a y = 4 Time (s) c. x = y = 87 d. x = 8 y = 7 L esso.2 Modelig Growth ad Decay 4 LESSON.2 Modelig Growth ad Decay 4