Section 4.6 Exercises

Transcription

1 Secion 4.6 Exponenial and Logarihmic Models 283 Secion 4.6 Exercises 1. You go o he docor and he gives you 13 milligrams of radioacive dye. Afer 12 minues, 4.75 milligrams of dye remain in your sysem. To leave he docor's office, you mus pass hrough a radiaion deecor wihou sounding he alarm. If he deecor will sound he alarm if more han 2 milligrams of he dye are in your sysem, how long will your visi o he docor ake, assuming you were given he dye as soon as you arrived? 2. You ake 200 milligrams of a headache medicine, and afer 4 hours, 120 milligrams remain in your sysem. If he effecs of he medicine wear off when less han 80 milligrams remain, when will you need o ake a second dose? 3. The half-life of Radium-226 is 1590 years. If a sample conains 200 mg, how many milligrams will remain afer 1000 years? 4. The half-life of Fermium-253 is 3 days. If a sample conains 100 mg, how many milligrams will remain afer 1 week? 5. The half-life of Erbium-165 is 10.4 hours. Afer 24 hours a sample has been reduced o a mass of 2 mg. Wha was he iniial mass of he sample, and how much will remain afer 3 days? 6. The half-life of Nobelium-259 is 58 minues. Afer 3 hours a sample has been reduced o a mass of 10 mg. Wha was he iniial mass of he sample, and how much will remain afer 8 hours? 7. A scienis begins wih 250 grams of a radioacive subsance. Afer 225 minues, he sample has decayed o 32 grams. Find he half-life of his subsance. 8. A scienis begins wih 20 grams of a radioacive subsance. Afer 7 days, he sample has decayed o 17 grams. Find he half-life of his subsance. 9. A wooden arifac from an archeological dig conains 60 percen of he carbon-14 ha is presen in living rees. How long ago was he arifac made? (he half-life of carbon-14 is 5730 years) 10. A wooden arifac from an archeological dig conains 15 percen of he carbon-14 ha is presen in living rees. How long ago was he arifac made? (he half-life of carbon-14 is 5730 years)

2 284 Chaper A baceria culure iniially conains 1500 baceria and doubles every half hour. Find he size of he populaion afer: a) 2 hours, b) 100 minues 12. A baceria culure iniially conains 2000 baceria and doubles every half hour. Find he size of he populaion afer: a) 3 hours, b) 80 minues 13. The coun of baceria in a culure was 800 afer 10 minues and 1800 afer 40 minues. a. Wha was he iniial size of he culure? b. Find he doubling period. c. Find he populaion afer 105 minues. d. When will he populaion reach 11000? 14. The coun of baceria in a culure was 600 afer 20 minues and 2000 afer 35 minues. a. Wha was he iniial size of he culure? b. Find he doubling period. c. Find he populaion afer 170 minues. d. When will he populaion reach 12000? 15. Find he ime required for an invesmen o double in value if invesed in an accoun paying 3% compounded quarerly. 16. Find he ime required for an invesmen o double in value if invesed in an accoun paying 4% compounded monhly The number of crysals ha have formed afer hours is given by n 20e How long does i ake he number of crysals o double?. 18. The number of building permis in Pasco years afer 1992 roughly followed he equaion n 400e. Wha is he doubling ime? 19. A urkey is pulled from he oven when he inernal emperaure is 165 Fahrenhei, and is allowed o cool in a 75 room. If he emperaure of he urkey is 145 afer half an hour, a. Wha will he emperaure be afer 50 minues? b. How long will i ake he urkey o cool o 110?

3 Secion 4.6 Exponenial and Logarihmic Models A cup of coffee is poured a 190 Fahrenhei, and is allowed o cool in a 70 room. If he emperaure of he coffee is 170 afer half an hour, a. Wha will he emperaure be afer 70 minues? b. How long will i ake he coffee o cool o 120? 21. The populaion of fish in a farm-socked lake afer years could be modeled by he 1000 equaion P e a. Skech a graph of his equaion b. Wha is he iniial populaion of fish? c. Wha will he populaion be afer 2 years? d. How long will i ake for he populaion o reach 900? 22. The number of people in a own ha have heard rumor afer days can be modeled by 500 he equaion N e. a. Skech a graph of his equaion b. How many people sared he rumor? c. How many people have heard he rumor afer 3 days?? d. How long will i ake 300 people o have heard he rumor? Find he value of he number shown on each logarihmic scale Plo each se of approximae values on a logarihmic scale Inensiy of sounds: Whisper: 10 W / m, Vacuum: 10 W / m, Je: 10 W / m 28. Mass: Amoeba: 5 10 g, Human: 5 10 g, Saue of Libery: 8 10 g The 1906 San Francisco earhquake had a magniude of 7.9 on he MMS scale. A he same ime here was an earhquake wih magniude 4.7 ha caused only minor damage. How many imes more inense was he San Francisco earhquake han he second one?

4 286 Chaper The 1906 San Francisco earhquake had a magniude of 7.9 on he MMS scale. A he same ime here was an earhquake wih magniude 6.5 ha caused less damage. How many imes more inense was he San Francisco earhquake han he second one? 31. One earhquake has magniude 3.9. If a second earhquake has 750 imes as much energy as he firs, find he magniude of he second quake. 32. One earhquake has magniude 4.8. If a second earhquake has 1200 imes as much energy as he firs, find he magniude of he second quake. 33. A colony of yeas cells is esimaed o conain 10 6 cells a ime = 0. Afer collecing experimenal daa in he lab, you decide ha he oal populaion of cells a ime hours is given by he funcion f 10 e [UW] a. How many cells are presen afer one hour? b. How long does i ake of he populaion o double?. c. Cherie, anoher member of your lab, looks a your noebook and says:...ha formula is wrong, my calculaions predic he formula for he number of yeas 6 cells is given by he funcion f Should you be worried by Cherie s remark? d. Anja, a hird member of your lab working wih he same yeas cells, ook 6 6 hese wo measuremens: cells afer 4 hours; cells afer 6 hours. Should you be worried by Anja s resuls? If Anja s measuremens are correc, does your model over esimae or under esimae he number of yeas cells a ime? 34. As ligh from he surface peneraes waer, is inensiy is diminished. In he clear waers of he Caribbean, he inensiy is decreased by 15 percen for every 3 meers of deph. Thus, he inensiy will have he form of a general exponenial funcion. [UW] a. If he inensiy of ligh a he waer s surface is I 0, find a formula for I( d ), he inensiy of ligh a a deph of d meers. Your formula should depend on I0 and d. b. A wha deph will he ligh inensiy be decreased o 1% of is surface inensiy?

5 Secion 4.6 Exponenial and Logarihmic Models Myoglobin and hemoglobin are oxygen carrying molecules in he human body. Hemoglobin is found inside red blood cells, which flow from he lungs o he muscles hrough he bloodsream. Myoglobin is found in muscle cells. The funcion p Y M p calculaes he fracion of myoglobin sauraed wih oxygen a a 1 p given pressure p orrs. For example, a a pressure of 1 orr, M(1) = 0.5, which means half of he myoglobin (i.e. 50%) is oxygen sauraed. (Noe: More precisely, you need o use somehing called he parial pressure, bu he disincion is no imporan for his problem.) Likewise, he funcion Y H p 2.8 p 26 p of hemoglobin sauraed wih oxygen a a given pressure p. [UW] a. The graphs of M ( p ) and H ( p ) are given here on he domain 0 p 100; which is which? b. If he pressure in he lungs is 100 orrs, wha is he level of oxygen sauraion of he hemoglobin in he lungs? calculaes he fracion c. The pressure in an acive muscle is 20 orrs. Wha is he level of oxygen sauraion of myoglobin in an acive muscle? Wha is he level of hemoglobin in an acive muscle? d. Define he efficiency of oxygen ranspor a a given pressure p o be M ( p) H ( p). Wha is he oxygen ranspor efficiency a 20 orrs? A 40 orrs? A 60 orrs? Skech he graph of M ( p) H ( p) ; are here condiions under which ranspor efficiency is maximized (explain)? 36. The lengh of some fish are modeled by a von Beralanffy growh funcion. For 0.18 Pacific halibu, his funcion has he form L e where L( ) is he lengh (in cenimeers) of a fish years old. [UW] a. Wha is he lengh of a new-born halibu a birh? b. Use he formula o esimae he lengh of a 6 year old halibu. c. A wha age would you expec he halibu o be 120 cm long? d. Wha is he pracical (physical) significance of he number 200 in he formula for L( )?

6 288 Chaper A cancerous cell lacks normal biological growh regulaion and can divide coninuously. Suppose a single mouse skin cell is cancerous and is mioic cell cycle (he ime for he cell o divide once) is 20 hours. The number of cells a ime grows according o an exponenial model. [UW] a. Find a formula C( ) for he number of cancerous skin cells afer hours. b. Assume a ypical mouse skin cell is spherical of radius cm. Find he combined volume of all cancerous skin cells afer hours. When will he volume of cancerous cells be 1 cm 3? 38. A ship embarked on a long voyage. A he sar of he voyage, here were 500 ans in he cargo hold of he ship. One week ino he voyage, here were 800 ans. Suppose he populaion of ans is an exponenial funcion of ime. [UW] a. How long did i ake he populaion o double? b. How long did i ake he populaion o riple? c. When were here be 10,000 ans on board? d. There also was an exponenially-growing populaion of aneaers on board. A he sar of he voyage here were 17 aneaers, and he populaion of aneaers doubled every 2.8 weeks. How long ino he voyage were here 200 ans per aneaer? 39. The populaions of ermies and spiders in a cerain house are growing exponenially. The house conains 100 ermies he day you move in. Afer 4 days, he house conains 200 ermies. Three days afer moving in, here are wo imes as many ermies as spiders. Eigh days afer moving in, here were four imes as many ermies as spiders. How long (in days) does i ake he populaion of spiders o riple? [UW]