Learning Enhancemen Team Model answers: Exponenial Funcions Exponenial Funcions sudy guide 1 i) The base rae of growh b is equal o 3 You can see his by noicing ha 1b 36 in his sysem, dividing boh sides by 1 gives b 3 ii) The saring number A is equal o 4 You can see his by noicing when 1 hen Ab 1 1 As b 3, you can subsiue his value ino he formula and rearrange o ge A 4 iii) The general exponenial funcion ha describes his sysem is y 3 4 In his sysem, C 34, D 916, E 8748 and he value of he sysem when 15 is 5739568 You can wrie down he general exponenial funcion by puing in he values you found for A and b ino he general formula for an exponenial sysem k y Ab here ha k 1 as he sysem grows a base rae b once every ime sep You know You can see ha a ime 4, he sysem is C So you can wrie C 4 3 4 4 81 34 You can see ha a ime 6, he sysem is D So you can wrie D 4 3 6 4 79 916 You can see ha a ime 7, he sysem is E So you can wrie E 4 3 7 4 187 8748 You can see ha a ime 15, he value of he sysem is some number y So you can wrie:
y 43 15 5739568 iv) I akes 7 6 minues for he sysem o grow above 9 Remember he logarihmic ransformaion from he sudy guide: y A k y b log b k A You are given in his case ha y 9 You can pu his value in o he general exponenial formula for his sysem o ge 9 4 3 Using he logarihmic ransformaion gives: 9 4 3 log 9 4 3 9 You can see here ha is equal o log 3 7 6 4 o 3 dp i) The base rae of decay b is equal o 5 8 You can see his by noicing ha 16 b he equaion o ge b 5 in his sysem, and so you can rearrange ii) The general exponenial funcion ha describes his sysem is y 5 In his sysem, F, G 4, H 3, I 64, J 18 and he value of he sysem when 1 is 14 You can wrie down he general exponenial funcion by puing in he values you found for A and b ino he general formula k y Ab for an exponenial sysem As he sysem is decaying once by he base rae a each ime sep, you know here ha k 1 You can see ha a ime 1, he sysem is F So you can wrie F 5 1 5
You can see ha a ime, he sysem is G So you can wrie G 5 5 4 You can see ha a ime 5, he sysem is H So you can wrie H 5 5 315 3 You can see ha a ime 6, he sysem is I So you can wrie I 5 6 1565 64 You can see ha a ime 7, he sysem is J So you can wrie J 5 7 7815 18 You can see ha a ime 1, he value of he sysem is some number y So you can wrie y 5 1 976565 14 iv) I akes 8 584 minues for he sysem o decay below 1 Remember he logarihmic ransformaion from he sudy guide: y A k y b log b k A You are given in his case ha y 1 You can pu his value in o he general exponenial formula for his sysem o ge 1 5 Using he logarihmic ransformaion gives: 1 1 5 log 5
1 You can see here ha log 5 8 584 o 3 dp and so 8 584 minues 3 Maerial (s) (s -1 ) (s) carbon-15 6 5 4 83 449 oxygen-15 45359 5 4 6 14 beryllium-11 5 5 13863 5 13863 dubnium-61 15971 5 44 6 666 nobelium-53 15 5 1984 7 99 : Remember he wo equaions given in he workshee: e (1) 1/ () You will need boh o complee he able For carbon-15, you are given ha 5, 4 s and 83 s-1 You can pu hese values ino (1) o ge: 5 e 83 4 6 To work ou you can pu your value 83 s -1 ino () o ge: 1/ 449 s 83 For oxygen-15, you are given ha 45 359, 4 s and 1 4 s You can pu your value of ino () o ge 1/ 14 You can rearrange his o ge 6 s -1 14
ow you know he value of, you can pu his and your values for ge: and ino (1) o 45539 6 4 e You can rearrange his o ge: 45539 5 (6 4 ) e For beryllium-11, you are given ha 5, 5 and 5 s -1 You can pu hese values ino (1) o ge: 5 5 e 5 You can use he logarihmic ransformaion o ge: 5 5 e 5 5 ln 5 5 You can rearrange his o ge 1 ln 13863 s 5 To work ou you can pu your value 5 s -1 ino () o ge: 1/ 13863 s 5 As was he ime aken o lose half of he number of aoms (as 5 is half of 5), in his case For dubnium-61, you are given ha 5, 159 71 and 44 s You can pu hese values ino (1) o ge: 15971 5 e 44 You can use he logarihmic ransformaion o ge: 159 71 e 44 5 15971 ln 44 5
You can rearrange his o ge: ln 15971 5 44 6 To work ou you can pu your value 6 s -1 ino () o ge: 1/ 666 s 6 For nobelium-53, you are given ha 15, 198 4 s and 99 s You can pu your value of ino () o ge 99 You can rearrange his o ge 7 s -1 99 ow you know he value of, you can pu his and your values for ge: and ino (1) o 15 7 198 4 e You can rearrange his o ge: 15 5 7 198 e 15 4 These model answers are one of a series on mahemaics produced by he Learning Enhancemen Team wih funding from he UEA Alumni Fund Scan he QR-code wih a smarphone app for more resources