. Notice how the initial amount is irrelevant when solving for half-life.

Transcription

1 7 Chaper 4 ecion 4.6 Exponenial and Logarihmic Models While we have explored some basic applicaions of exponenial and arihmic funcions, in his secion we explore some imporan applicaions in more deph. Radioacive Decay In an earlier secion, we discussed radioacive decay he idea ha radioacive isoopes change over ime. One of he common erms associaed wih radioacive decay is halflife. Half Life The half-life of a radioacive isoope is he ime i akes for half he subsance o decay. Given he basic exponenial growh/decay equaion h solving for when half he original amoun remains; by solving simply 1 b ( ) ab, half-life can be found by 1 a a( b), or more. Noice how he iniial amoun is irrelevan when solving for half-life. Example 1 Bismuh-1 is an isoope ha decays by abou 1% each day. Wha is he half-life of Bismuh-1? We were no given a saring quaniy, so we could eiher make up a value or use an unknown consan o represen he saring amoun. To show ha saring quaniy does no affec he resul, le us denoe he iniial quaniy by he consan a. Then he decay d of Bismuh-1 can be described by he equaion Q ( d) a(.87). To find he half-life, we wan o deermine when he remaining quaniy is half he original: 1 a. olving, 1 d a a(.87) Dividing by a, 1 d.87 Take he of boh sides 1 d (.87 ) Use he exponen propery of s 1 d (.87) Divide o solve for d

2 ecion 4.6 Exponenial and Logarihmic Models 71 1 d days.87 ( ) This ells us ha he half-life of Bismuh-1 is approximaely 5 days. Example Cesium-17 has a half-life of abou years. If you begin wih mg of cesium-17, how much will remain afer years? 6 years? 9 years? ince he half-life is years, afer years, half he original amoun, 1mg, will remain. Afer 6 years, anoher years have passed, so during ha second years, anoher half of he subsance will decay, leaving 5mg. Afer 9 years, anoher years have passed, so anoher half of he subsance will decay, leaving 5mg. Example Cesium-17 has a half-life of abou years. Find he annual decay rae. ince we are looking for an annual decay rae, we will use an equaion of he form Q ( ) a(1 + r). We know ha afer years, half he original amoun will remain. Using his informaion 1 a a(1 + r) Dividing by a 1 (1 + r) Taking he h roo of boh sides 1 1+ r ubracing one from boh sides, 1 r 1.84 This ells us cesium-17 is decaying a an annual rae of.84% per year. Try i Now Chlorine-6 is eliminaed from he body wih a bioical half-life of 1 days. Find he daily decay rae. hp://

3 7 Chaper 4 Example 4 Carbon-14 is a radioacive isoope ha is presen in organic maerials, and is commonly used for daing hisorical arifacs. Carbon-14 has a half-life of 57 years. If a bone fragmen is found ha conains % of is original carbon-14, how old is he bone? To find how old he bone is, we firs will need o find an equaion for he decay of he carbon-14. We could eiher use a coninuous or annual decay formula, bu op o use he coninuous decay formula since i is more common in scienific exs. The half life ells us ha afer 57 years, half he original subsance remains. olving for he rae, 1 r57 a ae Dividing by a 1 r57 e Taking he naural of boh sides 1 r57 ln ln( e ) Use he inverse propery of s on he righ side 1 ln 57r Divide by 57 1 ln r Now we know he decay will follow he equaion Q( ) ae. To find how old he bone fragmen is ha conains % of he original amoun, we solve for so ha Q().a..11.a ae. e. 11 ln(.) ln( e. 11 ) ln(.).11 ln(.) 11 years.11 The bone fragmen is abou 1, years old. Try i Now. In Example, we learned ha Cesium-17 has a half-life of abou years. If you begin wih mg of cesium-17, will i ake more or less han years unil only 1 milligram remains?

4 ecion 4.6 Exponenial and Logarihmic Models 7 Doubling Time For decaying quaniies, we asked how long i akes for half he subsance o decay. For growing quaniies we migh ask how long i akes for he quaniy o double. Doubling Time The doubling ime of a growing quaniy is he ime i akes for he quaniy o double. Given he basic exponenial growh equaion h solving for when he original quaniy has doubled; by solving ( ) ab, doubling ime can be found by x a a( b), or more x simply b. Again noice how he iniial amoun is irrelevan when solving for doubling ime. Example 5 Cancer cells someimes increase exponenially. If a cancerous growh conained cells las monh and 6 cells his monh, how long will i ake for he number of cancer cells o double? Defining o be ime in monhs, wih corresponding o his monh, we are given wo pieces of daa: his monh, (, 6), and las monh, (-1, ). From his daa, we can find an equaion for he growh. Using he form C ( ) ab, we know immediaely a 6, giving C( ) 6b. ubsiuing in (-1, ), 1 6b 6 b 6 b 1. This gives us he equaion C ( ) 6(1.) To find he doubling ime, we look for he ime unil we have wice he original amoun, so when C() a. a a(1.) (1.) ( ) ( 1. ) ( ) ( 1.) ( ).8 monhs. ( 1.) I will ake abou.8 monhs for he number of cancer cells o double.

5 74 Chaper 4 Example 6 Use of a new social neworking websie has been growing exponenially, wih he number of new members doubling every 5 monhs. If he sie currenly has 1, users and his rend coninues, how many users will he sie have in 1 year? We can use he doubling ime o find a funcion ha models he number of sie users, and hen use ha equaion o answer he quesion. While we could use an arbirary a as we have before for he iniial amoun, in his case, we know he iniial amoun was 1,. r If we use a coninuous growh equaion, i would look like N( ) 1e, measured in housands of users afer monhs. Based on he doubling ime, here would be 4 housand users afer 5 monhs. This allows us o solve for he coninuous growh rae: r5 4 1e r5 e ln 5r ln r Now ha we have an equaion, N( ) 1e, we can predic he number of users afer 1 monhs:.186(1) N (1) 1e 6.14 housand users. o afer 1 year, we would expec he sie o have around 6,14 users. Try i Now. If uiion a a college is increasing by 6.6% each year, how many years will i ake for uiion o double? Newon s Law of Cooling When a ho objec is lef in surrounding air ha is a a lower emperaure, he objec s emperaure will decrease exponenially, leveling off owards he surrounding air emperaure. This "leveling off" will correspond o a horizonal asympoe in he graph of he emperaure funcion. Unless he room emperaure is zero, his will correspond o a verical shif of he generic exponenial decay funcion.

6 ecion 4.6 Exponenial and Logarihmic Models 75 Newon s Law of Cooling The emperaure of an objec, T, in surrounding air wih emperaure T s will behave according o he formula k T ( ) ae + Ts Where is ime a is a consan deermined by he iniial emperaure of he objec k is a consan, he coninuous rae of cooling of he objec While an equaion of he form is more common. T ) ab + ( T could be used, he coninuous growh form s Example 7 A cheesecake is aken ou of he oven wih an ideal inernal emperaure of 165 degrees Fahrenhei, and is placed ino a 5 degree refrigeraor. Afer 1 minues, he cheesecake has cooled o 15 degrees. If you mus wai unil he cheesecake has cooled o 7 degrees before you ea i, how long will you have o wai? ince he surrounding air emperaure in he refrigeraor is 5 degrees, he cheesecake s emperaure will decay exponenially owards 5, following he equaion k T ( ) ae + 5 We know he iniial emperaure was 165, so T ( ) 165. ubsiuing in hese values, 165 ae k a + 5 a 1 We were given anoher pair of daa, T ( 1) 15, which we can use o solve for k k 15 1e k e 115 1k e ln 1k ln 1 k Togeher his gives us he equaion for cooling: T ( ) 1e + 5.

7 76 Chaper 4 Now we can solve for he ime i will ake for he emperaure o cool o 7 degrees e e. 1 5 e ln ln I will ake abou 17 minues, or one hour and 47 minues, for he cheesecake o cool. Of course, if you like your cheesecake served chilled, you d have o wai a bi longer. Try i Now 4. A picher of waer a 4 degrees Fahrenhei is placed ino a 7 degree room. One hour laer he emperaure has risen o 45 degrees. How long will i ake for he emperaure o rise o 6 degrees? Logarihmic cales For quaniies ha vary grealy in magniude, a sandard scale of measuremen is no always effecive, and uilizing arihms can make he values more manageable. For example, if he average disances from he sun o he major bodies in our solar sysem are lised, you see hey vary grealy. Plane Disance (millions of km) Mercury 58 Venus 18 Earh 15 Mars 8 Jupier 779 aurn 14 Uranus 88 Nepune 45 Placed on a linear scale one wih equally spaced values hese values ge bunched up. Mercury Venus Earh Mars Jupier aurn Uranus Nepune disance

8 ecion 4.6 Exponenial and Logarihmic Models 77 However, compuing he arihm of each value and ploing hese new values on a number line resuls in a more manageable graph, and makes he relaive disances more apparen. 4 Plane Disance (millions of km) (disance) Mercury Venus 18. Earh Mars 8.6 Jupier aurn Uranus Nepune Mercury Venus Earh Mars Jupier aurn Uranus Nepune (disance) omeimes, as shown above, he scale on a arihmic number line will show he values, bu more commonly he original values are lised as powers of 1, as shown below. P A B C D Example 8 Esimae he value of poin P on he scale above The poin P appears o be half way beween - and -1 in value, so if V is he value of his poin, ( V ) 1.5 Rewriing in exponenial form, V I is ineresing o noe he large gap beween Mars and Jupier on he number line. The aseroid bel, which scieniss believe consiss of he remnans of an ancien plane, is locaed here.

9 78 Chaper 4 Example 9 Place he number 6 on a arihmic scale. ince ( 6). 8, his poin would belong on he scale abou here: Try i Now 5. Plo he daa in he able below on a arihmic scale 5. ource of ound/noise Approximae ound Pressure in µpa (micro Pascals) Launching of he pace hule,,, Full ymphony Orchesra,, Diesel Freigh Train a High peed a 5 m, Normal Conversaion, of Whispering a m in Library, Unoccupied Broadcas udio ofes ound a human can hear Noice ha on he scale above Example 8, he visual disance on he scale beween poins A and B and beween C and D is he same. When looking a he values hese poins correspond o, noice B is en imes he value of A, and D is en imes he value of C. A visual linear difference beween poins corresponds o a relaive (raio) change beween he corresponding values. Logarihms are useful for showing hese relaive changes. For example, comparing $1,, o $1,, he firs is 1 imes larger han he second. 1,, 1 1 1, Likewise, comparing $1 o $1, he firs is 1 imes larger han he second. 1, When one quaniy is roughly en imes larger han anoher, we say i is one order of magniude larger. In boh cases described above, he firs number was wo orders of magniude larger han he second. 5 From hp:// rerieved Oc, 1

10 ecion 4.6 Exponenial and Logarihmic Models 79 Noice ha he order of magniude can be found as he common arihm of he raio of he quaniies. On he scale above, B is one order of magniude larger han A, and D is one order of magniude larger han C. Orders of Magniude Given wo values A and B, o deermine how many orders of magniude A is greaer han B, A Difference in orders of magniude B Example 1 On he scale above Example 8, how many orders of magniude larger is C han B? The value B corresponds o 1 1 The value C corresponds o 1 5 1, 5 1, 1 The relaive change is 1 1. The of his value is. 1 1 C is hree orders of magniude greaer han B, which can be seen on he scale by he visual difference beween he poins on he scale. Try i Now 6. Using he able from Try i Now #5, wha is he difference of order of magniude beween he sofes sound a human can hear and he launching of he space shule? An example of a arihmic scale is he Momen Magniude cale (MM) used for earhquakes. This scale is commonly and misakenly called he Richer cale, which was a very similar scale succeeded by he MM. Momen Magniude cale For an earhquake wih seismic momen, a measuremen of earh movemen, he MM value, or magniude of he earhquake, is M 16 Where 1 is a baseline measure for he seismic momen.

11 8 Chaper 4 Example 11 If one earhquake has a MM magniude of 6., and anoher has a magniude of 8., how much more powerful (in erms of earh movemen) is he second earhquake? ince he firs earhquake has magniude 6., we can find he amoun of earh movemen. The value of is no paricularly relevan, so we will no replace i wih is value Doing he same wih he second earhquake wih a magniude of 8., From his, we can see ha his second value s earh movemen is 1 imes as large as he firs earhquake. Example 1 One earhquake has magniude of.. If a second earhquake has wice as much earh movemen as he firs earhquake, find he magniude of he second quake. ince he firs quake has magniude.,

12 ecion 4.6 Exponenial and Logarihmic Models ince he second earhquake has wice as much earh movemen, for he second quake, Finding he magniude, M ( ) M The second earhquake wih wice as much earh movemen will have a magniude of abou.. In fac, using properies, we could show ha whenever he earh movemen doubles, he magniude will increase by abou.1: M + () M + () M +.1 M This illusraes he mos imporan feaure of a scale: ha muliplying he quaniy being considered will add o he scale value, and vice versa.

13 8 Chaper 4 Imporan Topics of his ecion Radioacive decay Half life Doubling ime Newon s law of cooling Logarihmic cales Orders of Magniude Momen Magniude scale Try i Now Answers 1 1. r or 6.7% is he daily rae of decay.. Less han years, o be exac. I will ake years, or approximaely 11 years, for uiion o double hours 5. Broadcas Conversaion ofes room of ound Whisper Train ymphony pace hule x The sound pressure in µpa creaed by launching he space shule is 8 1 x1 orders of magniude greaer han he sound pressure in µpa creaed by he sofes sound a human ear can hear.

14 ecion 4.6 Exponenial and Logarihmic Models 8 ecion 4.6 Exercises 1. You go o he docor and he injecs you wih 1 milligrams of radioacive dye. Afer 1 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 whenever more han 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 and he amoun of dye decays exponenially?. You ake milligrams of a headache medicine, and afer 4 hours, 1 milligrams remain in your sysem. If he effecs of he medicine wear off when less han 8 milligrams remain, when will you need o ake a second dose, assuming he amoun of medicine in your sysem decays exponenially?. The half-life of Radium-6 is 159 years. If a sample iniially conains mg, how many milligrams will remain afer 1 years? 4. The half-life of Fermium-5 is days. If a sample iniially conains 1 mg, how many milligrams will remain afer 1 week? 5. The half-life of Erbium-165 is 1.4 hours. Afer 4 hours a sample sill conains mg. Wha was he iniial mass of he sample, and how much will remain afer anoher days? 6. The half-life of Nobelium-59 is 58 minues. Afer hours a sample sill conains1 mg. Wha was he iniial mass of he sample, and how much will remain afer anoher 8 hours? 7. A scienis begins wih 5 grams of a radioacive subsance. Afer 5 minues, he sample has decayed o grams. Find he half-life of his subsance. 8. A scienis begins wih 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 archeoical dig conains 6 percen of he carbon-14 ha is presen in living rees. How long ago was he arifac made? (The half-life of carbon-14 is 57 years.)

15 84 Chaper 4 1. A wooden arifac from an archeoical dig conains 15 percen of he carbon-14 ha is presen in living rees. How long ago was he arifac made? (The half-life of carbon-14 is 57 years.) 11. A baceria culure iniially conains 15 baceria and doubles in size every half hour. Find he size of he populaion afer: a) hours b) 1 minues 1. A baceria culure iniially conains baceria and doubles in size every half hour. Find he size of he populaion afer: a) hours b) 8 minues 1. The coun of baceria in a culure was 8 afer 1 minues and 18 afer 4 minues. a. Wha was he iniial size of he culure? b. Find he doubling ime. c. Find he populaion afer 15 minues. d. When will he populaion reach 11? 14. The coun of baceria in a culure was 6 afer minues and afer 5 minues. a. Wha was he iniial size of he culure? b. Find he doubling ime. c. Find he populaion afer 17 minues. d. When will he populaion reach 1? 15. Find he ime required for an invesmen o double in value if invesed in an accoun paying % compounded quarerly. 16. Find he ime required for an invesmen o double in value if invesed in an accoun paying 4% compounded monhly e. 17. The number of crysals ha have formed afer hours is given by ( ).1 How long does i ake he number of crysals o double? 18. The number of building permis in Pasco years afer 199 roughly followed he.14 equaion n ( ) 4e. Wha is he doubling ime? n

16 ecion 4.6 Exponenial and Logarihmic Models 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 5 minues? b. How long will i ake he urkey o cool o 11?. A cup of coffee is poured a 19 Fahrenhei, and is allowed o cool in a 7 room. If he emperaure of he coffee is 17 afer half an hour, a. Wha will he emperaure be afer 7 minues? b. How long will i ake he coffee o cool o 1? 1. The populaion of fish in a farm-socked lake afer years could be modeled by he 1 equaion P( ) e a. kech a graph of his equaion. b. Wha is he iniial populaion of fish? c. Wha will he populaion be afer years? d. How long will i ake for he populaion o reach 9?. The number of people in a own who have heard a rumor afer days can be modeled 5 by he equaion N( ) e. + a. kech a graph of his equaion. b. How many people sared he rumor? c. How many people have heard he rumor afer days? d. How long will i ake unil people have heard he rumor? Find he value of he number shown on each arihmic scale Plo each se of approximae values on a arihmic scale Inensiy of sounds: Whisper: 1 W / m, Vacuum: 1 W / m, Je: 1 W / m 8. Mass: Amoeba: 5 1 g, Human: 5 1 g, aue of Libery: 8 1 g

17 86 Chaper 4 9. The 196 an Francisco earhquake had a magniude of 7.9 on he MM scale. Laer here was an earhquake wih magniude 4.7 ha caused only minor damage. How many imes more inense was he an Francisco earhquake han he second one?. The 196 an Francisco earhquake had a magniude of 7.9 on he MM scale. Laer here was an earhquake wih magniude 6.5 ha caused less damage. How many imes more inense was he an Francisco earhquake han he second one? 1. One earhquake has magniude.9 on he MM scale. If a second earhquake has 75 imes as much energy as he firs, find he magniude of he second quake.. One earhquake has magniude 4.8 on he MM scale. If a second earhquake has 1 imes as much energy as he firs, find he magniude of he second quake.. A colony of yeas cells is esimaed o conain 1 6 cells a ime. Afer collecing experimenal daa in he lab, you decide ha he oal populaion of cells a ime hours is given by he funcion f ( ) 1 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: Tha formula is wrong, my calculaions predic he formula for he number of yeas 6 cells is given by he funcion. ( ) 1 (.477 ) f. hould 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. hould 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? 4. 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 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, find a formula for Id, ( ) he inensiy of ligh a a deph of d meers. Your formula should depend on I and d. b. A wha deph will he ligh inensiy be decreased o 1% of is surface inensiy?

18 ecion 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 Torrs. For example, a a pressure of 1 Torr, M(1).5, which means half of he myoglobin (i.e. 5%) is oxygen sauraed. (Noe: More precisely, you need o use somehing called he parial pressure, bu he disincion is no imporan for.8 p his problem.) Likewise, he funcion Y H( p) calculaes he fracion 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 p 1; which is which? b. If he pressure in he lungs is 1 Torrs, wha is he level of oxygen sauraion of he hemoglobin in he lungs? c. The pressure in an acive muscle is Torrs. 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 Torrs? A 4 Torrs? A 6 Torrs? kech he graph of M( p) H( p) ; are here condiions under which ranspor efficiency is maximized (explain)? 6. The lengh of some fish are modeled by a von Beralanffy growh funcion. For.18 L 1.957e where L () is Pacific halibu, his funcion has he form ( ) ( ) he lengh (in cenimeers) of a fish years old. [UW] a. Wha is he lengh of a newborn 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 1 cm long? d. Wha is he pracical (physical) significance of he number in he formula for L ()?

19 88 Chaper 4 7. A cancer cell lacks normal bioical growh regulaion and can divide coninuously. uppose a single mouse skin cell is cancerous and is mioic cell cycle (he ime for he cell o divide once) is 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? 8. A ship embarked on a long voyage. A he sar of he voyage, here were 5 ans in he cargo hold of he ship. One week ino he voyage, here were 8 ans. uppose 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 1, 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.8 weeks. How long ino he voyage were here ans per aneaer? 9. The populaions of ermies and spiders in a cerain house are growing exponenially. The house conains 1 ermies he day you move in. Afer 4 days, he house conains 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]