hw-4_solutions

hw-4_solutions - Sec 3.2 12 First we equate the...

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Sec 3.2 12. First, we equate the expression (21) for 0 p at time a t and b t , we get 1 1 1 0 1 (1 ) a a Ap t a Ap t a p p e p p p e - - = - - and 1 1 1 0 1 (1 ) b b Ap t b Ap t b p p e p p p e - - = - - . Since 2 b a t t = , we set 1 a Ap t e χ - = and 1 2 b Ap t e - = , hence 2 1 1 2 1 1 (1 ) (1 ) a b a b p p p p p p p p = - - - - 1 1 ( ) ( ) a b b a p p p p p p - = - . Then substituting 1 1 ( ) ( ) a b b a p p p p p p - = - into 1 0 1 (1 ) a a p p p p p = - - and solving for 1 p , 1 1 1 0 1 1 1 ( ) ( ) ( ) (1 ) ( ) a b a b a a b a b a p p p p p p p p p p p p p p p p p - - = - - - - 1 1 0 1 1 1 1 ( ) ( ) ( ( ) ( )) a a b b a a b a a b p p p p p p p p p p p p p p p p p - = - - - - - 2 1 0 2 1 ( ) 2 a b b a b a p p p p p p p p p - = - + 0 0 1 2 0 2 ( ) a b b a a a b p p p p p p p p p p p - + = - . Now we find A from 1 a Ap t e - = , i.e., from 1 1 1 ( ) ( ) a Ap t a b b a p p p e p p p - - = - , which yields: 1 1 1 ( ) 1 ln[ ] ( ) b a a a b p p p A p t p p p - = - 0 0 2 0 0 0 1 2 0 2 (( ) ) 1 ln[ ] 2 (( ) ) a b b a b a a a b a b b a a a a b a b p p p p p p p p p p p p p p p p p p p t p p p p p p - + - - = - + - - = 2 0 0 0 2 1 0 0 0 (( 2 ) ) 1 ln[ ] ( 2 ) ( ) b a b b a a b a a b b a a b a b p p p p p p p p p p p t p p p p p p p p p p p - + - - - + - - 0 1 0 ( ) 1 ln[ ] ( ) b a a b a p p p p t p p p - = - . 14. If we set 0 t = to be the year 1970, then substituting 0 300 p = into formula 0 ( ) kt p t p e = we have: ( ) 300 kt p t e = . (*) We also know that by 1980 ( 10 t = years) the population had grown to 1500, thus
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10 (10) 300 1500 k p e = = ln 5 10 k = Substituting this value into equation (*), we have: ln5 10 ( ) 300 t p t e = . Hence, the population in 2010 ( 40 t = years) is ln5 40 4 10 (40) 300 300 5 187500 p e = = = . 15. Similarly, we set 0 t = to be the year 1970, then 0 300 p = . We also know that (5) 1200 p = (population in 1975) and (10) 1500 p = (population in 1980). Then we substitute (5) 1200 p = and (10) 1500 p = into the formula ( 29 1 0 1 0 1 0 ( ) Ap t p p p t p p p e - = + - , and obtain two nonlinear equations for the two unknowns 1 p and A : ( 29 1 1 5 1 300 1200 300 300 Ap p p e - = + - , ( 29 1 1 10 1 300 1500 300 300 Ap p p e - = + - . To solve
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hw-4_solutions - Sec 3.2 12 First we equate the...

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