Date: Worksheet 6-6: Exponential Growth and Decay Modelling. Doubling: Doubling time is the time it takes for a population to double in size.

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1 Dae: Workshee 6-6: Exponenial Growh and Decay Modelling Doubling: Doubling ime is he ime i akes for a populaion o double in size. The relaion for doubling is P P0 (2 ) d, where P represens he populaion, P 0 represens he iniial populaion, represens ime d represens he doubling ime, and he base 2 indicaes doubling 1. A baceria culure began wih 7500 baceria. Is growh can be modelled using he formula N 7500(2) 36, where N is he number of baceria afer hours. (a) Wha is he doubling ime? (b) How many baceria are presen afer 36 hours? (c) How many baceria are presen afer 72 hours? How does his relae o he doubling ime? Assigned Work: WS 6-7; Texbook p. 331 #4-7, #10

2 Dae: Half-Life: Half-life is he ime i akes for a quaniy o decay o half is original amoun. The relaion for doubling is h M M 0, where M represens he final quaniy, M 0 represens he iniial quaniy, represens ime h represens he half-life, and he base 2 1 indicaes half-life 2. All living organisms conain a known concenraion of 1 par per rillion pars of carbon-14. Carbon-14 is a radioacive elemen. I is used o dae ancien arefacs because i has a half life of abou 5730 years afer he organism dies. The formula C is used o calculae he concenraion, C, in pars per rillion, remaining n years afer deah. n (a) Wha is he iniial concenraion of carbon-14 as given in he formula n 5730 C? (b) Wha would be he concenraion of carbon-14 in a piece of cloh (made from plan fibres) afer 5730 years? (c) Wha would be he concenraion of carbon-14 in an animal bone afer years? Round your answer o five decimal places.

3 Dae: 3. E. coli is a very harmful ype of baceria ha can be found in mea ha is improperly sored or handled. The relaion N N 20 esimaes he number of E. coli, N, of an iniial sample of 0 0 ( 2) N baceria afer minues, a 37 C (body emperaure), under opimal condiions. (a) Wha is he doubling ime of E. coli? (b) If a sample of E. coli conains 5000 baceria, how many will here be afer 1 hour? (c) If a sample of E. coli conains 1000 baceria, how many will here be afer 1 day? 4. The deer populaion of a naional park was 250 deer 12 years ago. Today, here are 500 deer. Assuming he deer populaion has experienced exponenial growh, wrie a relaion represening he size of he deer populaion in he park. Use your relaion o projec he deer populaion in 25 years.

4 Dae: The relaion T 190 can be used o deermine he lengh of ime,, in hours, ha milk of a cerain fa conen will remain fresh. T is he sorage emperaure, in degrees Celsius. (a) Wha is he freshness half-life of milk? (b) How long will milk keep fresh a 22 C? (c) How long will milk keep fresh a 4 C?

5 Dae: 6. For his science projec, Ranji placed agar, a gel made from seaweed, in a peri dish and infeced i wih baceria. The measuremen of he growh ring was used o esimae he number of baceria presen. Ranji measured he growh ring for wo weeks. He consruced an exponenial growh model using he funcion N( ) 5(1.2) 2, where N () represens he number of baceria presen, in housands, and represens he ime, in days. (a) Wha is he number of baceria presen a he sar of he experimen? (b) How many baceria were presen afer one day? (c) How many baceria presen afer one week? (d) How long does i ake o double he number of baceria presen a he sar of he experimen? (e) Sae he domain and range of he funcion for he growh of his bacerial culure.

6 Dae: 7. Living organisms conain boh carbon-12, which does no decay, and radioacive carbon-14. When he organism dies, he amoun of carbon-14 decreases exponenially wih a half-life of abou 5730 years. One can measure he amoun of carbon-14 lef in an organism o esimae how long ago i died. The decay process can be modelled using he funcion P 100, where P represens he percen of carbon-14 remaining and represens he ime, in years. (a) Brislecone pine fossils found in he wesern Unied Saes show an age of abou 6000 years using carbon daing. Wha percen of he original carbon-14 remains in hese fossils? 5730 (b) Sedimens deposied by glaciers during he las ice age covered and buried layers of pea. Carbon daing shows abou 25% of he original carbon-14 remaining in he pea. Esimae he age of he glacier deposis o he neares hundred years. Answers: 1. (a) 36 hours, (b) , (c) , wo doubling periods in 72 hours, so baceria doubled wice; 2. (a) 1 par per rillion, (b) 0.5 pars per rillion, (c) pars per rillion; 3. (a) 20 min, (b) , (c) ; 4. (a) P 250(2) 12, (b) 2119; 5. (a) 10 hrs, (b) 31 hrs, (c) 56 hrs 6. (a) 5000 baceria, (b) 5477 baceria, (c) 9465 baceria, (d) abou 8 days, (e) D { x x R,0 x 14}, R { y y R,5 y 18} ; 7. (a) 48.4%, (b) abou years 24