Notes 8B Day 1 Doubling Time

Transcription

1 Noes 8B Day 1 Doubling ime Exponenial growh leads o repeaed doublings (see Graph in Noes 8A) and exponenial decay leads o repeaed halvings. In his uni we ll be convering beween growh (or decay) raes and doubling (or halving) imes. Doubling ime: he ime i akes for a quaniy ha is growing exponenially o double. Consider an iniial populaion of,000 ha grows wih a doubling ime of years. In years, or one doubling ime, he populaion increases by a facor of 2, o a new populaion of,000 x 2 =20,000. In 20 years, or 2 doubling imes, he populaion increases by a facor of 2 2, o a new populaion of,000 x 4 = 40,000. In 30 years, or 3 doubling imes, he populaion increases by a facor of 2 3, o a new populaion of,000 x 8 = 80,000. Calculaions wih Doubling ime Afer ime, he final amoun of a quaniy (ha s growing exponenially), wih a doubling ime of can be found by: y a 2 y=final amoun a=iniial amoun =ime ha has passed =doubling ime 2 = growh facor 1) Compound ineres produces exponenial growh because an ineres bearing accoun grows by he same percenage each year. Suppose your bank accoun has a doubling ime of 13 years. By wha facor does your balance increase in 50 years?

2 2) he world populaion doubled from 3 billion in 1960 o 6 billion in Suppose ha world populaion coninued o grow (from 2000 on) wih a doubling ime of 40 years. Wha would he populaion be in 2030? In 2200? 3) he number of ans in John s an farm is growing exponenially. A 8:00 AM he an farm had 1500 ans. en hours laer, he an populaion had doubled. a) If he an populaion coninues o grow a his same rae, how many ans will here be afer 1 week (from he 8:00 coun)? b) By wha facor will he an populaion increase in 24 hours? Consider an ecological sudy of a prairie dog communiy. he communiy conains 0 prairie dogs when he sudy begins, and researchers soon deermine ha he populaion is increasing a a rae of % per monh. ha is, each monh he populaion grows o 1% of, or 1.1 imes, is previous value. able 8.3 racks he populaion growh (rounded o he neares whole number). Use he able above o find he doubling ime for his prairie dog communiy.

3 Approximae Doubling ime Formula (Rule of 70) For a quaniy growing exponenially a a rae of P% per ime period, he doubling ime is approximaely 70 P% * Use he acual percen for P his approximaion works bes for small growh raes and breaks down for growh raes over abou 15%. Use he rule of 70 o find he doubling ime for his prairie dog communiy in he previous problem. 4) A own s populaion was abou 1.8 million in 2000 and was growing a a rae of abou 1.4% per year. Wha is he approximae doubling ime a his growh rae? By wha facor will he populaion increase in 80 years? 4.5) he number of weeds in my backyard is growing exponenially. In six days he number of weeds has doubled. Wha is he average percenage growh rae per day during his period?

4 Noes 8B Day 2 Exponenial Decay and Half-Life Exponenial decay occurs whenever a quaniy decreases by he same percenage in every fixed ime period (for example 20% every year).he amoun of he quaniy repeaedly decreases o half is amoun, wih each halving occurring in a ime called he half-life. Radioacive pluonium-239 (Pu-239) has a half-life of abou 24,000 years. Suppose 0- pound of Pu-239 is deposied a a nuclear wase sie. In 24,000 years, or one half-life, he amoun of Pu-239 declines o ½ he original amoun, or o (1/2) x 0 = 50 pounds In 48,000 years, or wo half-lives, he amoun of Pu-239 declines o (½) 2 he original amoun, or o (1/4) x 0 = 25 pounds In 72,000 years, or hree half-lives, he amoun of Pu-239 declines o (½) 3 he original amoun, or o (1/8) x 0 = 12.5 pounds Calculaions wih Half-Life Afer ime, he final amoun of a quaniy (ha s exponenially decaying), wih a half-life ime of can be found by: 1 y a 2 y=final amoun a=iniial amoun =ime ha has passed =half-life 1 2 = fracion of he iniial amoun ha remains Example 5 Carbon-14 Decay Radioacive carbon-14 has a half-life of abou 5700 years. I collecs in organisms only while hey are alive. Once hey are dead, i only decays. Wha fracion of he carbon-14 in an animal bone sill remains 00 years afer he animal had died?

5 Example 6 Pluonium Afer 0,000 Years Suppose ha 0 pounds of Pu-239 is deposied are a nuclear wase sie. How much of i will sill be presen in 0,000 years? Approximae Half-Life Formula (rule of 70) For a quaniy decaying exponenially a a rae of P% per ime period, he half-life is approximaely 70 P% * Use he acual percen for P his approximaion works bes for small decay raes and breaks down for decay raes over abou 15%. Example 7 Devaluaion of Currency Suppose ha inflaion causes he value of he Russian ruble o fall a a rae of 12% per year (relaive o he dollar). A his rae, approximaely how long does i ake for he ruble o lose half is value?

6 Exac Formulas for Doubling ime and Half-Life he approximae doubling ime and half-life formulas are useful because hey are easy o remember. However, for more precise work or for cases of larger raes where he approximae formulas break down, we need o exac formulas, given below. In Uni 9C, we will see how hey are derived. hese formulas use he fracional growh rae, defined as r = P/0, wih r posiive for growh and negaive for decay. For example, if he percenage growh rae is 5% per year, he fracional growh rae is r = 0.05 per year. For a 5% decay rae per year, he fracional growh rae is r = per year. Exac Doubling ime and Half-Life Formulas For an exponenially growing quaniy wih a fracional growh rae r, he doubling ime is double log 2 log (1 r) For an exponenially decaying quaniy, we use a negaive value for r (for example, if he decay rae is P = 5% per year, we se r = per year). he half-life is half log 2 log (1 r) Noe ha he unis of ime used for and r mus be he same. For example, if he fracional growh rae is 0.05 per monh, hen he doubling ime will also be measured in monhs. Also noe ha he formulas ensure ha boh double and half have posiive values. Example 8 Large Growh Rae A populaion of ras is growing a a rae of 80% per monh. Find he exac doubling ime for his growh rae and compare i o he doubling ime found wih he approximae doubling ime formula. Example 9 Ruble Revisied Suppose he Russian ruble is falling in value agains he dollar a 12% per year. Using he exac half-life formula, deermine how long i akes he ruble o lose half is value. Compare your answer o he approximae answer found in Example 7.