Section 5.8 Notes Page Exponential Growth and Decay Models; Newton s Law

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1 Sectin 5.8 Ntes Page Expnential Grwth and Decay Mdels; Newtn s Law There are many applicatins t expnential functins that we will fcus n in this sectin. First let s lk at the expnential mdel. Expnential Grwth / Decay Mdel A = kt ( A e A ( = current ppulatin, A = riginal ppulatin, k = grwth r decay cnstant t = time measured in any unit. EXAMPLE: At the start f the experiment there are 125 cells. Three hurs later there are 235 cells. a.) What is the expnential grwth frmula? b.) Hw many cells are present after 5 hurs? c.) Hw lng (runded t the nearest hur) will it take the ppulatin t reach 442 cells? We are ging t use the expnential grwth mdel abve. First we need t identify what infrmatin we are given. Since there are 125 cells at the start f the experiment we knw A = 125. It says after three hurs, s we knw that t = 3. It says that after 3 hurs there are 235 cells, s A ( = 235 since this is a current ppulatin. The nly thing we dn t knw is the grwth cnstant k. We need t find this. First we will substitute the infrmatin we identified. Part A. 235 = 125e 3k We have substituted the infrmatin s nw we need t slve fr k. 3k e = First divide bth sides by = e 3k We need t get rid f the e s we will take the natural lg f bth sides. ln 1.88 = ln e 3k The ln e frm the right side will cancel leaving us with 3k. ln 1.88 = 3k Nw divide bth sides by 3. ln1.88 k = Yu can always rund k t 4 decimal places. 3 Nte, if yu decide t use the whle number fr k instead f runding, this is kay. If yu dn t rund anything then yu might get slightly different answers than I gt, but yu will be clse enugh. Yu will still be crrect if yu are within a few decimal places. We are ready t answer part a. This asks us fr the grwth functin t When we write this we DO NOT put anything in fr t r A(. Our answer t part a is A( = 125e Nw fr part B. It asks fr hw many cells are present after 5 hurs. We knw that t = 5. We will put this int (5) ur grwth frmula: A ( 5) = 125e. T slve this we need t first multiply the tw numbers in the expnent: A ( 5) = 125e. T slve this we need t find ut what e is. If yu have a graphing calculatr, hit the secnd key and then the ln key t get e^. Then enter If yu have a scientific calculatr yu will need t enter first and then hit the secnd key and then ln. Either way yu shuld get Nw enter this int ur frmula: A ( = 125( ) 358 cells. Yu can rund it t the nearest cell.

2 Nw fr Part C. The 442 is the A (. We knw everything except the t. Sectin 5.8 Ntes Page = 125e t First divide bth sides by = e t Take the natural lg f bth sides. ln = ln e t The ln e cancels, leaving yu with t. ln = t Divide bth sides by t 6 hurs. EXAMPLE: At the start f the experiment there are 1000 bacteria. After 4 hurs the ppulatin dubled. What is the expnential grwth frmula? Hw many bacteria are present after 6 hurs? Since the ppulatin dubled, we knw that the new ppulatin will be 2000, therefre we knw A = The time is NOT 6 hurs. We need t use the time it tk the ppulatin t duble, which is 4 hurs, s t = 4. At the start f the experiment there are 1000 bacteria, s A ( = We need t put this int the frmula and slve fr k: k First divide bth sides by = e 4k Nw take the natural lg f bth sides. ln 2 = ln e 4k The ln e will cancel, leaving us with 4k n the right side. ln 2 = 4k Divide bth sides by 4 and rund t 4 decimal places. k t Nw we need t knw the grwth frmula fr this prblem. It is: A(. We need t knw the (6) ppulatin after 6 hurs, s t = 6. We will put this int ur grwth frmula: A (. T slve this we need t first multiply the tw numbers in the expnent: A (. T slve this we need t find ut what e is first by using ur calculatr. Yu will get Nw enter this int ur frmula: A ( = 1000( ) 2828 bacteria. EXAMPLE: Cmplete the table: 2000 pp Prjected 2011 Prjected Grwth (in millins) Pp (millins) Rate Let s lk at the table and define sme given variables. Since we are ging frm 2000 t 2011, we knw that t = 11. We are given the grwth rate, s we knw k = We als knw the initial ppulatin in 2000, s A = We just need t find the final ppulatin. Let s plug in ur given values int the grwth frmula (11) and use ur calculatr: A ( = 53.4e = 53.4e = 53.4( ) = millin.

3 Half-life Prblems Sectin 5.8 Ntes Page 3 The next prblems will fcus n types f decay. We will first lk at half-life prblems. Half-life is the amunt f time it takes half f a substance t decay. The prblems we will d will cntain this infrmatin. We will still use the same expnential mdel, except that half-life prblems have a special frmula t slve fr k. Decay cnstant fr a half-life prblem: k ln 2 =. half life EXAMPLE: The half-life f strntium-90 is 28 years. What is the decay functin? Hw much f a 10 gram sample is left after 11 years? We will nt use the 90 in this prblem. This is just the name f the istpe. We are given the half-life is 28 ln 2 years, s we can put this int the frmula t find ur k: k = Our k value shuld be 28 negative since we are wrking with a decay prblem. We knw that the initial amunt is 10 grams. Here is ur t decay functin: A(. Ntice whenever we find the grwth functin we d nt put anything in fr t r A( (11) T answer the secnd questin, we will nw put in a 11 fr t: A (. First multiply the tw numbers in the expnent: A (. Then we d the e part n ur calculatr and yu shuld nw have: A( = 10( ) = grams. Yu can rund yur answer t 2 decimal places. EXAMPLE: The half-life f radium-226 is 1620 years. Hw much f a 2 gram sample is left after 1000 years? We will nt use the 226 in this prblem. This is just the name f the istpe. We are given the half-life is 1620 ln 2 years, s we can put this int the frmula t find ur k: k = E 4. Yur scientific 1620 calculatr may display what I put abve. If it has an E next t it then this means the calculatr is displaying the answer in scientific ntatin. Yu need t mve the decimal 4 places t the left, s k = We knw that t the initial amunt is 2 grams. Here is ur decay functin: A(. T answer the secnd questin, we will nw put in a 1000 fr t: numbers in the expnent: A ( 0.4 A( = 2( ) = grams. A ( (1000). First multiply the tw. Then we d the e part n ur calculatr and yu shuld nw have: EXAMPLE: An artifact fund cntains apprximately 1.23% f the riginal amunt f carbn-14. The half life f carbn-14 is 5600 years. Apprximate the age f the artifact. ln 2 We can find the grwth cnstant: k = = We are nt given an amunt fr A. We are nly 5600 given that we have 1.23% f the riginal amunt left ver. If the riginal amunt is A, then 1.23% f this is A. This will be ur current ppulatin, s A( = A. Nw we substitute:

4 0.0123A 0 N t = A e Divide bth sides by Sectin 5.8 Ntes Page = e t Nw take the natural lg f bth sides ln ln = ln e t The ln e will cancel = t Divide bth sides by t years This will be the apprximate age f the artifact. Other types f decay prblems include Newtn s Law f Cling: Newtn s Law f Cling: kt = T + ( u T e, where k < 0. u ( ) u ( = final temperature after cling T = temperature f the surrunding atmsphere u = initial temperature befre cling k = cling cnstant (must be negative) t = time EXAMPLE: A pizza remved frm the ven has a temperature f 450 F. It is left sitting in a rm that has a temperature f 70 F. After 5 minutes the temperature f the pizza is 300 F. What is the temperature f the pizza after 20 minutes? When will the temperature f the pizza be 140 F? Let s first define ur variables. The initial temperature, u is 450 F. The temperature f the rm is the atmsphere s temperature, s T = 70 F. The time, t, is 5 minutes. The final temperature after cling is 300 F, s this is u (. We have everything we need t slve fr the cling cnstant, k. 300 = 70 + (450 70)e 5k We substitute int the frmula. 300 = e 5k We simplify inside the parenthesis. Nw subtract 70 frm bth sides. 230 = 380e 5k Divide bth sides by = e 5k Take the natural lg f bth sides ln = ln e 5k ln = 5k Divide bth sides by 5. k We have ur k value runded t 4 decimal places t Our cling frmula is nw: u( = 70 + (450 70) e, r questin we need t put in 20 fr t. Yu will get: u( t = +. Nw t answer the first e (20) u (20) = e First simplify multiply in the expnent u = + e Nw calculate (20) e. u ( 20) = ( ) Nw simplify t get the answer. u( 20) 121 F

5 Sectin 5.8 Ntes Page 5 T answer the secnd questin, nw we are given the final temperature after cling, u (, which is 140 F. = e t This time we need t slve fr t. Subtract 70 frm bth sides. 380e t = e 1004t Take the natural lg f bth sides. = ln e 1004t = Divide bth sides by ln ln = t Divide bth sides by t 17 min EXAMPLE: A thermmeter reading 72 F is placed in a refrigeratr where the temperature is a cnstant 38 F. After 2 minutes the thermmeter reads 60 F. What will it read after 7 minutes? Hw lng will it take befre thermmeter reads 39 F? Let s first define ur variables. The initial temperature, u is 72 F. The temperature f the refrigeratr is the atmsphere s temperature, s T = 38 F. The time, t, is 2 minutes. The final temperature after cling is 60 F, s this is u (. We have everything we need t slve fr the cling cnstant, k. 60 = 38 + (72 38)e 2k We substitute int the frmula. 60 = e 2k We simplify inside the parenthesis. Nw subtract 38 frm bth sides. 22 = 34e 5k Divide bth sides by = e 2k Take the natural lg f bth sides ln = ln e 2k ln = 2k Divide bth sides by 2. k We have ur k value runded t 4 decimal places t Our cling frmula is nw: u( = 38 + (72 38) e, r questin we need t put in 7 fr t. Yu will get: u( t = e. Nw t answer the first (7) u (7) = e First simplify multiply in the expnent u (7) = e Nw calculate e. u ( 7) = ( ) Nw simplify t get the answer. u( 7) F T answer the secnd questin, nw we are given the final temperature after cling, u (, which is 140 F. 39 = e t This time we need t slve fr t. Subtract 38 frm bth sides. 1 = 34e t Divide bth sides by = e 2177t Take the natural lg f bth sides. ln = ln e 02177t ln = t Divide bth sides by t 16.2 min We will be skipping the material cvering Lgistic Grwth Mdels and Mdeling Data.

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