Name: Algebra II Review for Quiz #13 Exponential and Logarithmic Functions including Modeling


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1 Name: Algebra II Review for Quiz #13 Exponenial and Logarihmic Funcions including Modeling TOPICS: Solving Exponenial Equaions (The Mehod of Common Bases) Solving Exponenial Equaions (Using Logarihms) The Number e and The Naural Logarihm Exponenial Growh vs. Decay Models Exponenial and Logarihmic Models (e.g. Compound Ineres, HalfLife, Newon s Law of Cooling, Logarihmic Models) Solve, round o he neares hundredh as needed () x (1.03) x 3. 3x e 4. ln(3x 4) x x 1 6. x Algebraically, deermine he x coordinae of he inersecion poin of he wo exponenial funcions shown below. 1 y 4 y 64 x x 1
2 8. Wrie as a single logarihm. 9. Expand. 3 1 y ln 40 ln ln x ln xz 10. For each of he following coordinae pairs, find he equaion of he exponenial funcion, in he form your answer. x y a b ha passes hrough he pair. Show he work ha you use o arrive a (a) 0,10 and 3, 80 (b) 0,180 and, Which of he following gives he savings S in an accoun if $50 was invesed a an ineres rae of 3% per year? (1) S 50 4 (3) S () S (4) S Sae he muliplier (base) you would need o muliply by in order o decrease a quaniy by he given percen lised. (a) 10% (b) % (c) 5% (d) 0.5% 13. If he populaion of a own is decreasing by 4% per year and sared wih 1,500 residens, which of he following is is projeced populaion in 10 years? Show he exponenial model you use o solve his problem. (1) 9,30 () 76 (3) 18,503 (4) 8,310
3 14. The sock price of WindpowerInc is increasing a a rae of 4% per week. Is iniial value was $0 per share. On he oher hand, he sock price in GerbilEnergy is crashing (losing value) a a rae of 11% per week. If is price was $10 per share when Windpower was a $0, afer how many weeks will he sock prices be he same? Model boh sock prices using exponenial funcions. Then, find when he sock prices will be equal graphically. Draw a well labeled graph o jusify your soluion. 15. If $130 is invesed in a savings accoun ha earns 4% ineres per year, which of he following is closes o he amoun in he accoun a he end of 10 years? (1) $18 (3) $168 () $19 (4) $ A populaion of 50 frui flies is increasing a a rae of 6% per day. Which of he following is closes o he number of days i will ake for he frui fly populaion o double? (1) 18 (3) 1 () 6 (4) If a radioacive subsance is quickly decaying a a rae of 13% per hour approximaely how much of a 00 pound sample remains afer one day? (1) 7.1 pounds (3) 5.6 pounds ().3 pounds (4) 15.6 pounds 18. Which of he following equaions would model a populaion wih an iniial size of 65 ha is growing a an annual rae of 8.5%? P (3) P (1) () P (4) P
4 19. The acceleraion of an objec falling hrough he air will decrease a a rae of 15% per second due o air resisance. If he iniial acceleraion due o graviy is 9.8 meers per second per second, which of he following equaions bes models he acceleraion seconds afer he objec begins falling? (1) a (3) a () a 15 (4) a Red Hook has a populaion of 6,00 people and is growing a a rae of 8% per year. Rhinebeck has a populaion of 8,750 and is growing a a rae of 6% per year. In how many years, o he neares year, will Red Hook have a greaer populaion han Rhinebeck? Show he equaion or inequaliy you are solving and solve i graphically. 1. A warm glass of waer, iniially a 10 degrees Fahrenhei, is placed in a refrigeraor a 34 degrees Fahrenhei and is emperaure is seen o decrease according o he exponenial funcion h T h (a) Verify ha he emperaure sars a 10 degrees Fahrenhei by evaluaing T 0. (b) Using your calculaor, skech a graph of T below for all values of h on he inerval 0 h 4. Be sure o label your yaxis and y inercep. (c) Afer how many hours will he emperaure be a 50 degrees Fahrenhei? Sae your answer o he neares hundredh of an hour. Illusrae your answer on he graph your drew in (b).. Percens combine in srange ways ha don' seem o make sense a firs. I would seem ha if a populaion grows by 5% per year for 10 years, hen i should grow in oal by 50% over a decade. Bu his isn' rue. Sar wih a populaion of 100. If i grows a 5% per year for 10 years, wha is is populaion afer 10 years? Wha percen growh does his represen?
5 3. A quaniy is growing a a consan 3% yearly rae. Which of he following would be is percen growh afer 15 years? (1) 45% (3) 56% () 5% (4) 63% 4. If a credi card company charges 13.5% yearly ineres, which of he following calculaions would be used in he process of calculaing he monhly ineres rae? (1) (3) () (4) The couny deb is growing a an annual rae of 3.5%. Wha percen rae is i growing a per years? Per 5 years? Per decade? Show he calculaions ha lead o each answer. Round each o he neares enh of a percen. 6. A populaion of llamas is growing a a consan yearly rae of 6%. A wha rae is he llama populaion growing per monh? Please assume all monhs are equally sized and ha here are 1 of hese per year. Round o he neares enh of a percen. 7. Shana is rying o increase he number of calories she burns by 5% per day. By wha percen is she rying o increase per week? Round o he neares enh of a percen. 8. If a bank accoun doubles in size every 5 years, hen by wha percen does i grow afer only 3 years? Round o he neares enh of a percen. Hin: Firs wrie an expression ha would calculae is growh rae afer a single year.
6 9. An objec s speed decreases by 5% for each minue ha i is slowing down. Which of he following is closes o he percen ha is speed will decrease over halfan hour? (1) 1% (3) 48% () 79% (4) 150% 30. Over he las 10 years, he price of corn has decreased by 5% per bushel. (a) Assuming a seady percen decrease, by wha percen does i decrease each year? Round o he neares enh of a percen. (b) Assuming his percen coninues, by wha percen will he price of corn decrease by afer 50 years? Show he calculaion ha leads o your answer. Round o he neares percen. 31. Which of he following represens he value of an invesmen wih a principal of $1500 wih an ineres rae of.5% compounded monhly afer 5 years? (1) $1, (3) $4,178. () $1, (4) $5, Franco invess $4,500 in an accoun ha earns a 3.8% ineres rae compounded coninuously. If he wihdraws he profi from he invesmen afer 5 years, how much has he earned on his invesmen? (1) $858.9 (3) $9.50 () $91.59 (4) $ An invesmen of $500 is made a.8% nominal ineres compounded quarerly. (a) Wrie an equaion ha models he amoun A he invesmen is worh years afer he principal has been invesed. (b) How much is he invesmen worh afer 10 years? (c) Algebraically deermine he number of years i will ake for he invesmen o be reach a worh of $800. Round o he neares hundredh.
7 ANSWERS: 1. x x x x x x 1 7. x 5 8. ln(10 x ) 9. 1 x ln y ln x ln z 10. (a) y 10() x (b) y () 1. (a)0.90 (b)0.98 (c) 0.75 (d) (4) weeks 15. () 16. (3) 17. (1) 18. () 19. (4) years 1. (a) T(0) = 10. Abou , abou 63% growh (b) Graph (c) 9.03 hours 3. (3) 4. (4) %, 18.8%, 41.1% % % % 9. () 30. (a).8% (b) 76% 31. () 3. (4) 33. (a).08 A (b) $660.9 (c) years 4
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