Pure Math 30: Explained!

Transcription

1 ure Mah : Explaied! 6

2 Logarihms Lesso ar Basic Expoeial Applicaios Expoeial Growh & Decay: Siuaios followig his ype of chage ca be modeled usig he formula: (b) A = Fuure Amou A o = iial Amou b = Type of Growh = eriod for Growh To Occur = Elapsed Time B & Examples A baceria doubles i hours (b =, = ) A radioacive sample has a half life of years (b = ½, = ) A populaio riples every years. (b =, = ) Example : A bacerial culure doubles every hours. f he culure sared wih baceria, how may baceria will be prese i hours? Example : The half life of a radioacive sample is hours. f 6 g of he sample was iiially prese, how much will remai afer 7 hours? Example : The populaio of a ow riples every 6 years. f people are prese i 6, how may will be i he ow i 6? 6 A = ( ) A = 6 A =76 baceria A = 96 people 7. g Quesios: ) A bacerial culure doubles every miues. f he culure sared wih baceria, how may baceria are prese afer miues? 7 ) The half life of a radioacive sample is. years. f 6 g of he sample was iiially prese, how much will remai afer 9 years? ) The populaio of a ow doubles every years. f people were prese i 99, how may people will be i he ow i? Aswers: ) 9 baceria ). g ) 6 people ure Mah : Explaied! 7

3 Logarihms Lesso ar Solvig For Oher Variables Example : A bacerial culure riples every hours. f he culure sared wih baceria, ad here are afer hours, wha is he value of i hours? =.6 = =. hours 6 - Graph he followig: Y =.6 Y= ^(/x) 6 7 A easy way o solve for variables i hese quesios is o graph ad fid he poi of iersecio. The equaios should be simplified as much as possible before graphig so you do have o worry abou fidig huge umbers i your widow seigs. algebraic mahods ivolvig log rules also work o solve hese equaios. The mehod you use is a maer of preferece. Example : The half life of a radioacive sample is 6. hours. f g of he sample is prese afer 7 hours, how much was iiially prese? 6. = A = A (.7) A = 7 g 7 Example : The populaio of a ow chages by a expoeial growh facor b every years. f people grows o 7 i years, wha is he value of b? 7 = b.977 = b b= This quesio is easily solved by basic algebra, so here is o eed o graph. Graph he followig: Y =.977 Y = x^(/) Quesios: ) A bacerial culure doubles every hours. f he culure sared wih baceria, ad here are afer hours, wha is he value of i hours? ) The half life of a radioacive sample is years. f g of he sample is prese afer years, how much was iiially prese? *You ca also use logarihm rules o obai he soluio! Algebraic soluio wih logarihms: Ab (.977) 7 = b.977 = b b =.9 = b ) The populaio of a ow chages by a expoeial growh facor b every years. f a populaio of 7 people icreases o a populaio of i year, wha is he value of b? Aswers: ) hour ).7 g ).6 ure Mah : Explaied!

4 Logarihms Lesso ar Raios of iial & Fial Amous Example : A bacerial culure doubles every hours. How log will i ake for a culure o quadruple? A = A = = hours f you are o give acual umbers for he iiial ad fuure values, you ca do i algebraically. Keep he iiial amou as A ad he express he fuure value i erms of A. This will cacel ou he A,, allowig you o solve he quesio. his example, he fuure value is imes greaer ha he iiial amou, so wrie he fuure value as A. Algebraic soluio wih logarihms: Example : A radioacive sample has a half life of days. How log will i ake for oly / of he sample o remai? Algebraic soluio wih logarihms:. A =A = =9 hours Example : The populaio of a ow riples every years. How may years will i ake for he populaio o double? A = A = =. years Graph he followig: Y = Y= ^(x/) 6 Graph he followig: Y= / Y= (/)^(x/) 6 Graph he followig: Y = Y= ^(x/) Ab Ab A = A ( ) = ( ) log = log ( ) log= log log = log = Quesios: ) A bacerial culure doubles every hours. How log will i ake for a culure o riple? A = A = log = log log = log Cross Muliply Algebraic soluio wih logarihms: Ab A = A ( ) = ( ) log = log ( ) log = log ( ) log = log =. log = log log = log = 9 Cross muliply ad divide Cross muliply ad divide ) A radioacive sample has a half life of days. How log will i ake for oly /6 of he sample o remai? ) The populaio of a ow quadruples every years. How may years will i ake for he populaio o double? Aswers: ).7 hours ).7 g ) years ure Mah : Explaied! 9

5 Logarihms Lesso ar V erces of iial & Fial Amous Example : Ligh passig hrough murky waer reais ¾ of is iesiy for every mere of waer. A wha deph will he ligh iesiy be 6% of wha i is a he surface?.6a = A.6 = =.7 m Example : The populaio of a ow halves every years. how may years will % of he populaio have fled?.a = A. = =. years Example : A radioacive sample has a half life of years, ad has a iiial mass of 6 g. How log will i ake for he sample o lose g? 6 = 6. = =. years Graph he followig: Y =.6 Y= (/)^x 6 Graph he followig: Y =. Y= (/)^(x/) Graph he followig: Y =. Y= (/)^(x/) The fial amou is always wha you have remaiig. his example, he remaiig ligh iesiy is 6%, so you ca use.6a wihou ay problems. his quesio, he fuure amou will NOT be.a, sice ha is he amou los. The remaiig amou will be % of he populaio, so use.a o he lef side. The fial amou is simply Algebraic soluio wih logarihms: Ab.6 A = A.6 = log.6 = log log.6 = log log.6 = log =.7 Algebraic soluio wih logarihms: Quesios: ) Ligh passig hrough murky waer reais 6/7 of is iesiy of for every mere of waer. A wha deph will he ligh iesiy be 6% of wha i is a he surface? ) The populaio of a ow halves every years. how may years will 7% of he populaio have fled? 6 g g = 6 g. ) A radioacive sample has a half life of years, ad has a iiial mass of 9 g. How log will i ake for he sample o lose g? Ab. A =. = A Aswers: ). m ). years ). years log. = log log. = log log. = log =. Algebraic soluio wih logarihms: Ab 6 = 6(.). = (.) log. = log(.) log. = log (.) log. = log (.) =. ure Mah : Explaied!

6 Logarihms Lesso ar V Rae Quesios Example : A crack i a widow grows by. % every hour. f he crack sars a a legh of cm, how log will i be i hours? A =.. cm Use. for he b-value sice we are addig.% o op of he iiial value for every hour ha passes. b - Values f he rae is a icreasig perce, add i o. f he rae is a decreasig perce, subrac i from. Example : Ligh passig hrough murky waer loses % of i s iesiy for every mere of waer. A wha deph will he ligh iesiy be half of wha i is a he surface? A =A.7 = (.7 ) =.9 m Example : A ow loses.% of is populaio every year. How may years will i ake for a populaio of o become 7? 7 =.99.7 =.99 =6.6 years Graph he followig: Y =. Y= (.7)^x Graph he followig: Y =.7 Y= (.99)^x Use.7 for he b value, sice we are reaiig 7% of he ligh for every mere. Use.99 for he b-value sice we are reaiig 99. % of he populaio each year. Algebraic soluio wih logarihms: Ab. A = A (.7). = (.7) log. = log (.7) log. = log (.7) log. = log.7 =.9 Algebraic soluio wih logarihms: Ab 7 = (.99).7 = (.99) log.7 = log(.99) log.7 = log(.99) log.7 = log.99 = 6.6 Quesios: ) A crack i a widow grows by 6. % every hour. f he crack sars a a legh of.7 cm, how log will i be i hours? ) Ligh passig hrough murky waer loses 6% of i s iesiy for every mere of waer. A wha deph will he ligh iesiy be / of wha i is a he surface? ) A ow loses.% of is populaio every year. How may years will i ake for a populaio of o drop uder 9? Aswers: ).76 cm ).6 m ). years ure Mah : Explaied!

7 Logarihms Lesso ar V Compoud eres Compoud eres: The formula for compoud ieres is A = A ( + i) A = Fuure Amou A = iial Amou = eres rae per compoudig period. (Divide aual ieres rae by period) = umber of compoudig periods. (Muliply years by period) Commo compoudig periods are: Aually: Oce per year Semi-Aually: Twice per year Quarerly: Four imes per year Mohly: imes per year Daily: 6 imes per year. Example : A sum of moey is ivesed a 6%, compouded quarerly, for years. Wha are he values required for i &? i = 6% =.% =. = x = Example : A sum of moey is ivesed a.7%, compouded semi-aually, for years. Wha are he values required for i &? i =.7% =.% =. = x = 6 Example : $ is ivesed a 7.% compouded aually for years. Wha is he amou of moey a he ed of he years? i = 7.% = 7.% =.7 = x = A = ( +.7) = $66. Example : $ is ivesed a 6% compouded mohly for 7 years. How much ieres is eared? i = 6% =.% =. = 7 x = A = ( +.) = $96. eres eared = $96. $ = $96. ure Mah : Explaied!

8 Logarihms Lesso ar V Compoud eres Example : $ is ivesed a % compouded quarerly. How may years mus i say i he bak o double? = % =.% =. = represes he A = ( +.) umber of compoudig 6 = (.) periods. = (.) log = log. log = log. log = = log. Quesios: Sice he compoudig is doe imes per year, ha meas he moey has bee i he bak for 7 years. ) $ is ivesed a.6% compouded semi-aually for 7 years. Wha is he amou of moey a he ed of he 7 years? ) $ is ivesed a 9.% compouded mohly for. years. How much ieres is eared? ) $ is ivesed a % compouded quarerly. How may years mus i say i he bak o riple? Aswers: ) $77.9 ) $76.9 ). years ure Mah : Explaied!

9 Logarihms Lesso ar V Earhquakes M-M earhquake iesiy: The formula for earhquakes is: = = esiy Raio of earhquakes M = Richer Magiude of Sroger Earhquake M = Richer Magiude of Weaker Earhquake The Richer Scale is a logarihmic way of measurig earhquake sregh. Example : a earhquake of magiude 6 is e imes sroger ha a magiude earhquake. Example : a earhquake of magiude 7 is imes sroger ha a magiude earhquake. Example : A small remor of magiude. is followed by a sroger oe of magiude.. How much sroger is he secod remor ha he firs? = = M-M.-. =. The larger earhquake is imes sroger ha he weaker. Example : A weak earhquake has a magiude of 6., ad he followig day a srog earhquake occurs wih double he iesiy. Wha is he magiude of he sroger earhquake? = M-M M-6. = M-6. log = log log = ( M - 6.) log log = ( M - 6.)( ) log = M - 6. M=log+6. M=6. Example : A srog earhquake wih a magiude of 6.7 is hree imes more iese ha a weaker earhquake. Wha is he magiude of he weaker earhquake? = M-M 6.7-M = 6.7-M log = log log = ( M ) log log = ( M )( ) log = M M =6.7-log M = 6. Quesios: ) A small remor of magiude. is he followed by a sroger oe of magiude.. How much sroger is he secod remor ha he firs? ) A earhquake has a magiude of 6.7, ad he followig week a sroger earhquake occurs wih six imes he iesiy. Wha is he magiude of he sroger earhquake? ) A earhquake wih a magiude of.9 is oe housad imes more iese ha a weaker earhquake. Wha is he magiude of he weaker earhquake? Aswers: ), ) 7. ).9 ure Mah : Explaied!

10 Logarihms Lesso ar V Soud Loudess Soud: The formula for soud iesiy is: db-d B = log db = Decible level of Loud Soud; dbb = Decible level of Sof Soud = esiy Raio of Souds Example : A soud of db is imes louder ha a weaker soud. Wha is he loudess of he weaker soud? db - db =log - db =log ( ) - db =6. db = -6. db = 6.7dB Example : Oe soud has a loudess of 7 db, ad aoher soud has a loudess of db. How much louder is he secod soud? db - db =log - 7 =log =log. = log. = =. The louder soud is. imes louder ha he weaker soud. Example : f he iesiy of oe soud is imes sroger ha aoher soud, wha is he differece i decibels bewee he wo souds? ΔdB =log ΔdB =log ΔdB = Use he " seve rule" The louder soud is db greaer ha he weaker soud. Quesios: ) A soud of 7 db is 9 imes louder ha weaker soud. Wha is he loudess of he weaker soud? ) Oe soud has a loudess of 97 db, ad aoher soud has a loudess of db. How may imes louder is he secod soud? ) f he iesiy of oe soud is imes sroger ha aoher soud, wha is he differece i decibels bewee he wo souds? Aswers: ). db ) ) 7 db ure Mah : Explaied!

11 Logarihms Lesso ar X Acids + ph: The formula for acid sregh is: ph = - log H ph = Acid Sregh (power of hydroge) H + = Coceraio of Hydroge o -6 Example : A beaker of acid has a hydroge coceraio of. mol/l Calculae he ph of he acid. + ph = -log H -6 ph = -log. ph =.6 The formula for fidig [H + ], give he ph, is H + = -ph Example : f a beaker of acid has a ph of., Calculae he hydroge coceraio of he acid. + -ph H = + -. H = Derivaio of above formula: + ph = - log H + - H = 7.9 mol/l Take he egaive o he oher side of he equaio o isolae he logarihm + - ph = log H Now Use he seve rule -ph = + H Example : A beaker of acid has a ph of.9, ad a secod beaker has a ph of 7.6. Deermie how may imes higher he hydroge coceraio is i he srog acid as compared o he weaker oe. Sep : Calculae + i he sroger acid H - H + = -.9 =.6 Sep : Calculae + i he weaker acid H - H + = -7.6 =. - srog.6 Sep : Divide he resuls: = = weak -. Quesios: - ) A beaker of acid has a hydroge coceraio of.9 mol/l Calculae he ph of he acid. ) f a beaker of acid has a ph of.7, Calculae he hydroge coceraio of he acid. ) A beaker of acid has a ph of., ad a secod beaker has a ph of.9. Deermie how much higher he hydroge coceraio is i he secod beaker as compared o he firs beaker. Aswers: ). ) -9 mol/l ) 9 imes sroger. ure Mah : Explaied! 6