Bartlett-Arithmetic+Growth.pdf

8/14/2019 Bartlett-Arithmetic+Growth.pdf

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360POPULATION AN D ENVIRONMENT

a nd e x p o n e n t i a l g r o w t h o r d e c a y are m o n o t o n i c . M o n o t o n i c g r o w t h o rd e c a y n e ed n o t b e e it h e r l i n e a r o r e x p o n e n t i a l .W e wa n t to discuss expon en t i a l changes, (g rowth and decay ) .W e w i l l d e f i n e t h e s y m b o l k t o b e t h e fr a c t io n a l c h a n g e p e r u n i t t im e

a n d t h e s y m b o l R t o b e t h e p e r c e n t ch a n g e p e r u n i t t im e w h i c h w e w i l lca l l t he g rowth ra te . Thus(1) R = 100 k

I f k = + 0 . 0 3 p e r y e a r, t h e n w e a re d e a l i n g w i t h a g r o w t h r a te o f R = 3percen t pe r year .I f k i s pos i t i ve and cons tan t , one has exponen t i a l g rowth ; i f k i s nega-

t i v e a n d c o n s t a n t, o n e h a s e x p o n e n t i a l d e c a y .I t is un fo r tuna te tha t t he te rm exp on en t i a l g row th is i n te rp re ted by

some to be a ra re and e xo t i c fo rm o f g row th tha t i s d i f f e ren t f rom thes t e a d y g r o w t h w h i c h is s o o f te n i n t h e n e w s . W e w i l l d e f i n e s t e a d y

g r o w t h t o b e s y n o n y m o u s w i t h e x p o n e n t ia l g r o w t h a n d g e o m e t ri cg r o w t h .

I t is i n c o r r e c t t o u se t h e t e rm l o g a r i t h m i c g r o w t h t o d e s c r ib e e x p o -nen t i a l g rowth . The book , The Logar i thm ic Ce ntury (Lapp, 1973) shouldhave been ca l l ed The Exp onen t ia l Cen tury. I t s h o w s m a n y e x a m p l e s o fq u a n t i t i e s t h a t h a v e b e e n g r o w i n g e x p o n e n t i a l l y f o r l o n g p e r i o d s o f t i m e .

T HE G R O W T H O F P O P U L T IO N SP o p u l a t io n s t e n d t o c h a n g e e x p o n e n t i a l l y . T h i s c a n b e s e e n f r o m t h e

uni ts that are used to express b i r th and death ra tes . The wor ld b i r th ra te i sa p p r o x i m a t e l y 2 7 p e r t h o u s a n d e a ch y e a r a n d t h e d e a th ra te i s a p p r o x -imate l y 10 per thousan d each year . The 27 p er thou san d expresses thef rac t i ona l chang e and the each yea r conve r t s th i s to the f rac t iona l changeper un i t t ime . The d i ff e rence be tw een these tw o num bers is t he inc rease o f17 per thousand each year , o r 1 .7 per hundred each year . Th i s means tha tthe g row th ra te is R = 1 .7 percen t pe r year , o r k = + 0 .0 17 per year .

S teady g row th occurs i n a per i od o f t ime i f t he va lue o f k is pos i t ivea n d c o n s t a n t t h r o u g h o u t t h e p e r i o d . V e r y o f t e n o n e d e a l s w i t h g r o w t h t h a tis c o n t i n u o u s b u t w h i c h is c h a r a c t e r iz e d b y a c h a n g i n g g r o w t h ra te . T h i scan be represented i n tw o w ays .a ) I f w e h a v e a c h a n g i n g g r o w t h r ate d u r i n g a p e r io d o f ti m e , t h egrow th m ay be represented by an average g ro wth rate w h ic h is charac -te r i zed by a cons tan t va lue o f k , o r


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361ALBERT A. BARTLETT

b) we can tabu la te a se r i es o f va lues o f k as k changes w i th t imet h r oug hou t t he pe r i od . I f k is pos i t iv e and i s i n c reas i ng w i t h t i m e , w e hav egrow th tha t i s fas te r than ex po ne n t i a l , and i f k is po s i t ive bu t dec reas ingw i t h t i m e , w e hav e g r ow t h t ha t i s s l ow e r t han ex po ne n t i a l I f k decreasesto ze ro and becomes nega t i ve , the changes sw i t ch f rom growth to decay .

I n add i t i on t o popu l a t i ons , t h i ngs t ha t t end t o g r ow ex ponen t i a l l y i n -c lude : m on ey i n an i n te res t -bear i ng sav ings accou n t , t he cos t o f l i v i ng , thenumber o f f i ss ion even ts tha t take p lace w i th each genera t i on o f neu t ronsi n a nuc l ea r ex p l os i on , t he num be r o f pages o f a r t i c l es pub l i s hed annua l l yin s c i en t i f ic j ou r na ls , and t he num be r o f k il om e t e rs o f h i gh w a y in theUn i ted S ta tes (Bar t le t t, 1969) . T h ings tha t tend to decay e xp on en t i a l l y i n -c l ude t he v a l ue o f t he do l l a r , t he num be r o f undec ay ed r ad i oac t i v e a t om sin a samp le , the amp l i t ude o f v i b ra t i ons o f osc i l l a t i ng ob jec ts , and thecharge on a capac i to r tha t i s d i scharg ing th rough a res i s to r .E X A M P L E N O 1

Here a re recen t da ta fo r the popu la t i on o f the Un i ted S ta tes1990 248 . 71 m i ll ion1 9 8 0 2 2 6 . 5 5 m i l l i o nInc rease i n the 1980s 22 .16 m i l l ion

The average o f the s ta r t i ng and end ing popu la t i ons o f the decade i s ,( 248 .71 + 226 . 55 ) / 2 = 237 . 63 m i l l ion

There a re th ree wa ys w e can express the f rac t i ona l inc rease i n the decade .( 2 2 .1 6 / 2 2 6 . 5 5 ) = 0 . 0 9 7 8 o r 9 . 7 8 %( 2 2 .1 6 / 2 3 7 . 6 3 ) = 0 . 0 9 3 3 o r 9 . 3 3 %( 2 2 .1 6 / 2 4 8 . 7 1 ) = 0 . 0 8 91 o r 8 . 9 t %

These resu l ts can be expressed by say ing tha t the po pu la t ion increase int he dec ade w as 9 . 78 % o f the p opu l a t ion a t t he s ta rt o f the dec ade ; t heinc rease was 9 .33 % o f the average o f the pop u la t i ons a t the s ta rt and theend o f t he dec ade ; o r the i nc r ease w as 8 . 91% o f t he pop u l a t ion a t t he endo f the decade . We can d i v i de these numbers by the ten years o f the decadet o ge t 0 . 97 8% , 0 . 93 3% , and 0 . 89 1% . W h i c h , i f any , o f these num be r s ist he av e r age annua l g r ow t h r a te f o r t he dec ade ? T o ans w e r t h is ques t i on ,w e m us t ex am i ne t he a r i t hm e t i c o f s t eady ( ex ponen t i a l ) g r ow t h .


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D IS R E TE O M P O U N D I N GThe a r i t hm et i c o f popu la t i on g row th is t he same as the a r i thm et i c o ft h e g r o w t h o f m o n e y in a s a vin g s a c c o u n t b e c a us e o f t h e c o m p o u n d i n te r -es t t ha t i s added to the accoun t . (A f te r one o f my ta l ks on th i s sub jec t , ab a n k e r in t h e a u d i e n c e s a i d t o m e t h a t h e w a s c o m p l e t e l y f a m i li a r w i t h t h eg r o w t h o f m o n e y d u e t o t h e a r i t h m e t i c o f c o m p o u n d i n t e r e s t , b u t h e h a dnever rea l i zed tha t t h i s a r i t hmet i c a l so app l i ed to the g row th o f popu la -t ions . )I f one leaves m one y un tou che d i n a sav ings acco un t , t he i n te res t isadded a t regu la r i n te rva l s and the i n te res t i s ca l cu la ted as a f i xed f rac t i on(say 5%) o f t he money i n the accoun t . Thus the f rac t i ona l g row th ra te , k , o fthe m on ey i n the acc oun t is a cons tan t 0 .05 pe r year , (5% per year) . Th i sg u a r a n t e e s t h a t t h e m o n e y i n t h e a c c o u n t w i l l g r o w s t e a d i l y ( e x p o n e n -t i a l l y ) . T h e s p e e d w i t h w h i c h t h e m o n e y g r o w s d e p e n d s o n t w o t h i n g s : t h ein te res t rate and the f requ enc y w i th w h i c h the i n te res t ca l cu la t i ons a remade . The imp or tanc e o f t he i n te res t rate is ob v iou s ; t he imp or tance o f t hef reque ncy o f com po un d in g (ca l cu la t i ng the i n te res t) is sma l l e r and is less

o b v i o u s .The e f fec t o f t he f requency o f compound ing i s i l l us t ra ted i n F ig .1 . Thef i gu re deals w i th 10 0 tha t is p l aced i n a sav ings acco un t fo r fi ve years a tan annua l i n te res t rate o f 1 2% . In F ig . l (a ) w e see the w ay i n w h i c h thedo l l a rs i ncrease w he n the 12% in te res t is ca l cu la ted ann ua l ly a t t he end o feach year . The number o f do l l a rs i nc reases i n a s tep fash ion , and the do l -l a rs i n one s tep a re found by mu l t i p l y i ng the do l l a rs on the p rev ious s tepb y 1 . 1 2 . T h e c o m p o u n d i n g d o n e a t t h e e n d o f th e fi ft h y e a r le a ve s17 6 .2 3 i n the acco un t . In F ig . l (b ) we s ta r t w i th the same 100 , bu t t he1 2 % a n n u a l in t e re s t is c o m p o u n d e d n o w a s 6 % t w i c e e a c h y e a r. T he re a r enow tw i ce as many s teps , and the do l l a rs i n one s tep a re found by mu l t i -p l y i n g t h e d o l l a r s o n t h e p r e v i o u s s t e p b y 1 . 0 6 . T h e c o m p o u n d i n g d o n e a tthe end o f t he f i ft h year leaves 17 9 .0 8 i n the acco un t . The 12% co u ld bec o m p o u n d e d a s 1 % e v e r y m o n t h . In t h is ca se , t h e a m o u n t a t t h e e n d o ft h e f if th y e a r is 1 8 1 . 6 7 . T h e 1 2 % c o u l d b e c o m p o u n d e d a t t h e d a i l y ra teo f ( t 2 / 3 6 5 ) % t o g i v e 1 8 2 . 1 9 a t t h e e n d o f f i v e y e ars .I f w e w a n t t o a p p l y c a l c u l a t io n s o f t h is t y p e to t h e c a l c u l a t io n o f th egrowth o f popu la t i ons , we see a t l eas t t h ree p rob lems .(1) Pop u la t ions do n o t g row s tepw ise as do the do l la rs i n F igs . l (a ) and(b);(2) The s i ze o f t he acc oun t a t t he end o f fi ve years depen ds on the fre -q u e n c y o f c o m p o u n d i n g , y e t w e d o n o t k n o w w h a t f re q u e n c y o f c o m -p o u n d i n g w o u l d b e a p p r o p r i a t e f o r u s e i n o u r p o p u l a t i o n c a l c u l a -t i ons ;


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FIGU RE 1 . Part (a ) show s the g row th o f 100 a t an annua l i n te res t rate o f1 2 % , c o m p o u n d e d o n c e a y e a r f o r fi v e y ea rs . P a r t (b ) s h o w s t h e g r o w t ho f 1 0 0 a t a n a n n u a l in t e re s t ra te o f 1 2 % c o m p o u n d e d t w i c e a y e a r f o rf i ve years . Par t (c) sho ws the gro w th o f 10 0 a t an annua l in teres t ra te o f12% com po un de d co n t i nu ou s l y fo r f ive years . Par t (c ) is t he g raph o fs t e a d y o r e x p o n e n t i a l g r o w t h .i~ ) L L A ~ S t l ' t o Y q I 5 i ' P ~ F 1 7 6 2 3

l l Z ' 1 2 5 J ' 1 i II 0 0 . o Ii

I 2 3 - f 5_ D O L L A R S

i 0 0 . o 0 l l Z . ~ O I ? . .6 .~5 ~ r r 1 5 0 . 3 6 1 6 g , ~ I ~, , 133~~ i 0 6 . o o I I ~ I . i

[I t

D o L L . ~ 5i , ? . . 2 s I Z i ' . I z

3 51 ~ 2 z t _161 61

0 I Z 3I T i M I y ~ R ~4


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(3) I t w o u l d b e d i f f ic u l t t o c a l c u l a t e t h e p o p u l a t i o n i n cr e a s e i n a n i n te r v a ls u c h a s 3 . 6 7 y e a r s i f w e w e r e c o m p o u n d i n g a n n u a l l y o r s e m i - a n n u -a l l y .

O N T I N U O U S O M P O U N D I N GA s o n e c a n s e e f ro m F i g . l ( a ) a n d (b ), t h e m o r e f re q u e n t th e c o m -

p o u n d i n g , t h e s m a l l e r a r e t h e s te p s i n t h e g r a p h t h a t re p r e s e n t s t h e g r o w t h .I n t h e l im i t , w e c a n c o m p o u n d c o n t i n u o u s l y , a n d th e ste ps b e c o m e s os m a l l t h a t th e y m e r g e i n t o a s m o o t h c u r v e . T h e e q u a t i o n f o r t h i s c u r v e is(2) N2 = N1 e k t2 - h)I n th i s e q u a t i o n ,

N 1 i s t h e s iz e o f t h e g r o w i n g q u a n t i t y a t t i m e h ,N 2 i s t h e s i z e o f t h e g r o w i n g q u a n t i t y a t t i m e t 2 ,

k is t h e f r a c t i o n a l i n c re a s e i n N p e r u n i t t im e , a n de = 2 . 7 1 8 2 8 . . . w h i c h is t h e b as e o f n a t u r a l l o g a r i t h m s .T h i s e q u a t i o n is m o s t o ft e n w r i tt e n i n th e f o r m

kt(3 ) N = No eI n t h i s f o r m , t h e g r o w i n g q u a n t i t y h a s t h e s i z e N a t t h e t i m e t , a n d i t h a dt h e s iz e N = N o a t t h e t i m e t = 0 . T h e q u a n t i t y t is a t i m e i n t e r v a l .

I t is i m p o r t a n t t h a t t h e u n i t s o f t i m e i n k a n d t b e th e s a m e . I f k h a s t h eu n i t s per month t h e n t h e u n i t s o f t m u s t b e months F i g u r e 1 ( c ) s h o w s t h eg r o w t h o f $ 1 0 0 i n a s a v in g s a c c o u n t a t a n a n n u a l i n te r e s t r at e o f 1 2 %c o m p o u n d e d c o n t i n u o u s l y f o r f i v e ye a rs . I t is c a l c u l a t e d f ro m E q .3 ,N = 1 0 0 e 12 t

w h e r e t i s t h e t i m e i n y e a rs . A t t h e e n d o f fi v e y e a r s , t h a s t h e v a l u e 5 , a n dN is c a l c u l a t e d t o h a v e th e v a l u e $ 1 8 2 . 2 1 .T o d o t he s e c a l c u l a t i o n s re q u i r e s a s m a l l h a n d - h e l d s c i e n t i f ic c a l c u -l a t o r t h a t h a s k e y s f o r t h e f u n c t i o n s u s e d b y s c ie n t i s ts a n d e n g i n e e r s . I np a r t i c u l a r t h e c a l c u l a t o r s h o u l d h a v e k e y s l a b e l e d e x ' , ,,y X ,,, I n x ( o rI n ) , a n d l o g x ( o r l o g ) . T h e s e c a l c u l a t o r s c a n b e p u r c h a s e d f o r a sl i t t l e as $20 .


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C o n t i n u o u s c o m p o u n d i n g e l i m i n a t e s t h e t h r e e p r o b l e m s t h a t a r i s ew i t h d i sc r et e c o m p o u n d i n g , s o w e w i l l u se t h e a ri th m e t ic o f c o n t i n u o u sc o m p o u n d i n g t o d e s c r i b e t h e e x p o n e n t i a l , o r s t e a d y , g r o w t h o f p o p u l a -t i o n s .

T H E V E R G E G R O W T H R TE

I f w e k n o w t h e p o p u l a t i o n N o a t t h e st a rt o f a n i n t e r v a l o f ti m e , a n dt h e p o p u l a t i o n N a t t h e e n d o f t h e i n t e r v a l , w e c a n d e f i n e t h e a v e r a g e r a t eo f g r o w t h i n t h e i n te r v a l as t h a t ra te o f g r o w t h , a n d t h e c o r r e s p o n d i n gc o n s t a n t v a l u e o f k , th a t le ts a q u a n t i t y o f s iz e N o g r o w t o th e s i z e N i n t h eg i v e n t i m e i n t e r v a l .

I f w e k n o w N a n d N o w e c a n c a l c u l a t e th e a v e ra g e g r o w t h ra te . T o d ot h i s , w e m u s t s o l v e E q . 3 f o r k .(4 ) k = (1 / t ) l n ( N / N o )

I n t h i s e q u a t i o n , l n ( N / N o ) m e a n s t h e n a t u r a l l o g a r i t h m o f th e q u o t i e n t Nd i v i d e d b y N o . W e c a n r e a r r a n g e E q . 4 t o f i n d t h e t i m e f o r N t o g r o w f r o mN o t o N f o r a g i v e n k .5 ) t = (1 / k ) l n ( N / N o )

B A C K T O E X A M P LE N O 1W e w i l l u s e E q . 4 t o c a l c u l a t e t h e a v e r a g e g r o w t h r a t e o f t h e p o p u l a -

t i o n o f t h e U n i t e d S ta te s i n t h e t i m e i n t e r v a l t = 1 0 y e a r s b e t w e e n 1 9 8 0a n d 1 9 9 0 .k = (1 / 10) In (248 .71 / 22 6 .5 5)k = (1 / 1 0 ) I n ( 1 .0 9 7 8 1 . . . )k = (1 / 1 0) x 0 . 0 9 3 3 2 1 8 . . .k = 0 . 0 0 9 3 3 2 1 8 . . . so th a t R = 0 . 9 3 3 . . % y e a r - 1

T h u s , a s t e a d y g r o w t h r a te o f 0 . 9 3 3 . . . % p e r y e a r , w i l l r e s u l t i n a p o p u l a -t i o n g r o w i n g fr o m 2 2 6 . 5 5 m i l l i o n t o 2 4 8 .7 1 m i l l i o n i n t en y e a rs . T o th r e es i g n i f i c a n t f ig u r e s , t h e a v e r a g e g r o w t h r a te h a p p e n s t o e q u a l o n e o f t h et h r e e g r o w t h r a t e s t h a t w e r e c a l c u l a t e d e a r l i e r .

F o r l i n e a r g r o w t h i n a n i n t e rv a l o f ti m e , t h e a v e ra g e o f t h e p o p u l a t i o n sa t t h e s t a rt a n d e n d o f t h e i n t e r v a l is t h e a v e r a g e p o p u l a t i o n d u r i n g t h e


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3 6 6

P O P U L A T I O N A N D E N V I R O N M E N T

i n te r v a l . F o r e x p o n e n t i a l g r o w t h , t h e a v e r a g e p o p u l a t i o n i n a n i n t e rv a l i s al i tt l e le ss t h a n t h e a v e r a g e o f t h e p o p u l a t i o n s a t th e s ta r t a n d e n d o f t h ei n t e r v a l . F o r s t e a d y g r o w t h f o r th e U n i t e d S ta te s f r o m 1 9 8 0 t o 1 9 9 0 , t h ea v e r a g e p o p u l a t i o n w a s 2 3 7 . 4 6 m i l l i o n .

T h e f o r m u l a f o r c a l c u l a t i n g t h e a v e ra g e p o p u l a t i o n , a s w e l l a s t h ek e y s tr o ke s n e e d e d t o m a k e s o m e o f th e s e c a l c u l a t io n s w i t h a s c i e n t if icc a l c u l a t o r a re g i v e n i n t h e A p p e n d i x .

T H E D O U B L I N G T I M EI f a q u a n t i t y is g r o w i n g 5 % p e r y e a r , i ts s i ze i s i n c r e a s i n g b y a f ix e d

f r a c t i o n ( 5 % ) i n a f i x e d l e n g t h o f ti m e ( o n e y e a r) , a n d t h i s is t r u e n o m a t te rw h e r e o n e is o n t h e g r o w t h c u r v e . I n d e e d , th i s is t h e c o n d i t i o n t h a t d e f in e ss t e a d y g r o w t h . I t t h e n f o l l o w s t h a t a l o n g e r fi x e d l e n g t h o f f t i m e is r e q u i r e df o r t h e g r o w i n g q u a n t i t y t o i n c r e a se its s i ze b y 1 0 0 % , w h i c h is a d o u b l i n go f i ts s iz e . T h i s l o n g e r t i m e is c a l l e d t h e dou l ing t ime and i t i s r ep r esen t edb y T 2. T h e d o u b l i n g t i m e is t h e t i m e r e q u i r e d fo r N t o g r o w f ro m its i n i ti a ls i ze No t o t he s i ze 2 No . F r om Eq . 3 ,

k T2)2 No = No ek T2)2 = e

I f w e t a k e t h e n a t u r a l l o g a r i t h m o f b o t h s i d e s ,

In 2 = k T 2(6) T2 = In 2 ) / k= 0 . 6 93 . . . / k , and s ince R = 100 k ,

= 10 0( .6 93 . . . ) / R= 6 9 . 3 / R ~ 7 0 / R

E q u a t io n (6 ) is k n o w n a s T h e L a w o f 7 0 . T h e e q u a l i ti e s o f E q .6 ar e e x a c tf o r c o n t i n u o u s c o m p o u n d i n g . N u m b e r s s l i g h t l y l ar g e r t h a n 6 9 . 3 a r e n e c e s -s a ry t o c a lc u l a te th e d o u b l i n g t i m e w h e n th e c o m p o u n d i n g is d o n e a n n u -a l l y o r s e m i a n n u a l l y . B a n k e rs s o m e t im e s c a l l t h is T h e L a w o f 7 2 .

T a b l e 1 s h o w s d o u b l i n g t i m e s f o r s e ve r al ra te s o f s te a d y g r o w t h , w h i l eT a b l e 2 s h o w s th e s i z e o f a g r o w i n g q u a n t i t y a ft e r s e v e r a l p e r i o d s o fg r o w t h , w h e r e t h e t im e s a r e e x p re s s e d in u n i ts i n t h e d o u b l i n g t im e .


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T A B L ED o u b l i n g T i m e s f o r D i ff e r e n t R a te s o f S t ea d y G r o w t h

Percent g row th Do ub l ing t imeper yea r in years

Zero Infinity0.5 139.01.0 69.31.5 46.22.0 34.73.0 23.14.0 17.35.0 13.910.0 6.9320.0 3.47

T A B L E 2S te ad y G r o w t h f o r D i f fe r e n t N u m b e r s o f D o u b l i n g T im e s

T ime, in numberso f doub l i ng t im es

S ize o f the g row ingquant i t y in mu l t ip lesof the in i t ia l s ize, No

Zero 122 43 84 165 32

1 1 24n

This concept o f the doubl ing t ime is appl icable for an increase in s izeby any fac tor . The t ime T6 for N to increase the s ize by a fac tor o f 6 is ,

T 6 = In 6 ) / k= 1 . 7 9 . . . / k- - 1 7 9 / R


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6 8

POPUL TION ND ENVIRONMENT

W h e n t h e q u a n t i t y N is d e c r e a s i n g i n s i ze b y a c o n s t a n t f r a c t io n p e ru n i t t i m e , k is n e g a t i v e , a n d t h e d o u b l i n g t i m e b e c o m e s t h e h a l f - l i f ew h i c h is t h e t im e f o r N t o d e c a y t o h a l f o f its in i t ia l v a l u e , N o . T h e d e c a y so f r a d i o a c t i v e m a t e r ia l s a r e c h a r a c t e r i z e d b y h a l f -l iv e s .

P OW E RS O F T W OT h e c o n c e p t o f t h e d o u b l i n g t im e a l lo w s u s t o m a k e a c o n v e n i e n t

r e f o r m u l a t i o n o f E q . 3 .(7) N = N o 2 (~T2/T h i s e q u a t i o n is v e r y u s e fu l b e c a u s e p o w e r s o f t w o a re e a s ie r t o c a l c u l a t ein o n e ' s h e a d t h a n p o w e r s o f e.E X A M P LE N O . 2 .B y w h a t f a c t o r d o e s a p o p u l a t i o n i n cr e a s e if i t g r o w s 5 p e r y e a r f o r 3 0years?

T h e d o u b l i n g t i m e is a p p r o x i m a t e l y ( 7 0 / 5 ) = 1 4 y e a r s .S o ( t / T 2 = 3 0 / 1 4 = 2 . 1 4 . . . d o u b l i n g t i me s .

T h e n N = No 2 2 .1 4 w h i c h is b e t w e e n 2 2 = 4 a n d 23 = 8 .W i t h a s c i e n t i fi c c a l c u l a t o r , No 2 2 1 4 = 4 . 4 . . . No . (T h e k e y s t r o k e s

f o r t h i s c a l c u l a t i o n a r e g i v e n i n t h e A p p e n d i x )T h u s th e f a c t o r w e a r e s e e k i n g is 4 . 4 . . .

E X P O NE N T I L G R O W T H F O R M N Y D O U B L I N G T IM ESA n i m p o r t a n t f e a tu r e o f s te a d y g r o w t h is t h a t a ft er l o n g p e r i o d s o f ti m e

( m a n y d o u b l i n g t im e s ) , t h e si ze o f t h e g r o w i n g q u a n t i t y b e c o m e s e n o r -m o u s .

F o r m e n t a l c a l c u l a t i o n s , i t is c o n v e n i e n t t o r e m e m b e r t h a t2 l = 1 0 2 4 ~ 1 03

S o t h e g r o w t h i n 2 5 d o u b l i n g t im e s c a n b e e s t im a t e d a s f o l l o w s :


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22s = 2 l x 2 l x 25 --~ 103 x 103 32 = 3 .2 x 107H e r e i s a n o t h e r c a l c u l a t i o n a l c o n v e n i e n c e . S te a d y g r o w t h o f n % p e r

y e a r f o r 6 9 . 3 y e a rs ( a p p r o x i m a t e l y o n e h u m a n l if e t im e i n t h e w e s t e r n i n -d u s t r i a l i z e d n a t i o n s ) r e s u lt s in a n o v e r a l l i n c r e a s e i n th e s i z e o f th e g r o w i n gq u a n t i t y b y a fa c t o r o f 2 n . F or e x a m p l e , I f a p o p u l a t i o n h a s s t ea d y g r o w t ho f 6 % p e r y e a r f o r 6 9 . 3 y e a r s , it s s i z e w i l l i n c re a s e b y a f a c t o r o f 26 = 6 4 .W h e r e o n e s c h o o l w a s n e e d e d a t t h e s ta rt o f th e p e r i o d , 6 4 s c h o o l s o f t h es a m e s iz e w i l l b e n e e d e d a t t h e e n d o f t h e p e r i o dEXAMPLE NO 3

T h e la rg e n u m b e r s th a t c o m e a s a c o n s e q u e n c e o f m a n y d o u b l in g s a r e th eb a s is f o r m a t h e m a t i c a l q u e s t i o n s s u c h a s t h is . Y o u a r e to w o r k f o r 3 0 d a y s .Y o u h a v e a c h o i c e o f a s a la r y o f $ 1 0 0 0 f o r a l l th i s w o r k , o r y o u c a n h a v e as a l a r y t h a t s t a r t s a t o n e c e n t t h e f i r s t d a y a n d g r o w s e x p o n e n t i a l l y , d o u -b l i n g e v e r y d a y f o r t h e 3 0 d a y s . W h i c h m e t h o d o f p a y m e n t w o u l d y o uprefer?

O n t h e f i r s t d a y y o u r s a l a r y , i n d o l l a r s , i sO n t h e s e c o n d d a yO n t h e t h i r d d a y2 / 1 0 0 = $ 0 .0 121 / 100 = $0 .022 2 / 1 0 0 = $ 0 . 0 4

O n t h e t h i r t i e t h d a y y o u r s a l a r y is 2 29 / 1 0 0w h i c h is $ 5 , 3 6 8 , 7 0 9 . 1 2I t c a n b e s h o w n t h a t th e t o t a l s a l a r y f o r t h e 3 0 d a y s i s ( 2 3 0 - 1 ) / 1 0 0 w h i c hi s $ 1 0 , 7 3 7 , 4 1 8 . 2 3 .

EXAMPLE NO 4E x c l u d i n g A n t a r c t i c a , t h e l a n d a re a o f t h e e a r t h is 1 . 2 4 x 1 0 1 4 s q u a r em e t e r s . I f t h e p o p u l a t i o n o f t h e e a r t h in 1 9 9 2 is 5 . 5 x 1 0 9 p e o p l e , a n d i f

t h is p o p u l a t io n c o n t i n u e s to g r o w s t e a d il y a t 1 .7 % p e r y e a r, w h e n w o u l dt h e p o p u l a t i o n d e n s i t y r e a c h o n e p e r s o n p e r s q u a r e m e t e r o n t h e d r y l a n ds u r f a c e o f t h e e a r t h ? W e c a n u s e E q . 5 .t = (1 / 0 . 0 1 7 ) 1 n ( 1. 24 x 1 0 1 4 / 5 . 5 x 1 09 )t = 5 9 0 y e a r s

S i n c e w e k n o w t h a t o n e p e r s o n p e r s q u a r e m e t e r i s a n i m p o s s i b l e p o p u l a -t i o n d e n s i ty , t h is a r i t h m e t i c t e l ls us t h a t w o r l d p o p u l a t i o n g r o w t h w i l l s t o pi n a t i m e s h o r t c o m p a r e d t o 5 9 0 y e a rs .


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37


EXAMPLE N O 5A s s um e t ha t t he w o r l d pop u l a t i on g r ow t h has been s t eady a t 1 . 7% pe r y ea rs inc e t he ti m e o f A dam and E v e . C a l c u la t e w hen t hey l i v ed . A ga i n w e useEq.5 . No w N = 5 .5 x 109 and No = 2 (Adam and Eve).

t = ( 1 / 0 . 0 1 7 ) 1 n (5 .5 x 1 0 9 / 2 )t = 1279 years ago , o r ab ou t 713 A .D .

Th i s es tab l i shes tha t the g rowth o f wor ld popu la t i on has no t been s teady ;over mos t o f recen t h i s to ry , t he g rowth has been faste r than expone nt ialTod ay ' s annua l g ro wth rate o f ap pro x im ate l y 1 .7% is mu ch la rger than theaverage g rowth ra te over a l l o f human h i s to ry [

T he annua l g r ow t h r ate o f w o r l d pop u l a t ion m ay hav e been 1 . 9% i nthe 1970s . I f t h i s is co r rec t , t hen the recen t pe r i od o f de c l i ne o f the po pu la -t i on g r ow t h r ate f rom 1 . 9% t o 1 . 7% is a pe r i od o f g r ow t h t ha t is s l ow e rt han ex p one n t i a l , i .e . , t he g r ow t h r ate i s dec l in i ng . G r ow t h t ha t is s l ow e rthan exp on en t i a l , and dec reas ing to k = 0 and R = 0 , is necessary if t heearth i s to reach ze ro po pu la t i on g row th .

G EN ER L C O N C L U S I O N S B O U T G R O W T H1 ) W hen ev e r a g r ow i ng qua n t i t y in cr eas es by a f ix ed f r ac ti on i n a f ix edper iod o f time , the g row th is s teady (expo nen t ia l ) .2 ) In a m odes t num be r o f do ub l i ng tim es , the g r ow i ng quan t i t y w i l l i n -c rease enormous l y i n s i ze .3 ) As a conse quen ce , the s ize o f th i ngs , o r the num ber o f th i ngs , cannev e r c on t i nue t o g r ow i nde f i n i t e l y .4 ) In a l l systems , g row th i s a shor t -te rm t rans ien t phe nom eno n.

T he em i nen t ec ono m i s t K enne t h B o u l d i ng s um m ed i t a ll up w hen heis r epo rt ed to hav e s a id , A n y on e w ho t h i nk s t ha t s t eady g r ow t h c an c on -t i nue i nde f i n i t e l y i s e i the r a m adm an , o r an ec o nom i s t .

The e f fec t o f s teady g ro wth i n the ra te o f consu m pt ion o f fi n i t e re -sources su ch as fossil fuels has been se t for th in deta i l (Bart let t, 1978).

Sus ta ined Y ie ld app l ies to agr icu l tura l resources and descr ibes a s i t -ua t i on i n wh ich the ra te o f use o f a rene wa b le resource equa l s the rate o fb io log i ca l regenera t ion th roug h p lan t g ro wth . Sus ta ined av a i l ab i l i t y is acon cep t tha t can be app l ied to nonre new ab le resources . For sus ta inedav a i l a b i l i t y , t he rate o f con sum pt ion o f a f i n i t e resource mus t have exp o-


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F I G U R E 2 . A t ti m e t = 0 w e h a v e t w o p o p u l a t i o n s o f 1 0 0 p e rs o n s e a c h .T h e l o w e r c u r v e r e p r e s e n ts a l in e a r g r o w t h o f 1 0 p e r s o n s p e r y e a r . T h eu p p e r c u r v e r e p r e s e n ts s t e a d y g r o w t h o f 1 0 p e r y e a r . T h e s lo p e o f t h ec u r v e s r e p r e s e n t t h e ra te o f c h a n g e o f e a c h p o p u l a t i o n . A t t im e t = 0 ,b o t h p o p u l a t i o n s a r e c h a n g i n g a t a ra te o f 1 0 p e o p l e p e r ye a r, b u tb e c a u s e o f t h e s te a d y g r o w t h , t h e r a te o f c h a n g e o f th e u p p e r c u r v ei n c re a s e s w i t h t i m e .I I I I

POPUL~T ON-1 6 0 ~ P= i~S

STE ADY / P 50

L I o 0 _ ~ U N F - A R . C T R O W T HT ~ M E

I 2 3 4 5 / E A R 5O I I I I In e n t i a l d e c a y a t a c e r t a i n r a t e , a n d t h i s w i l l a l l o w t h e r e s o u r c e t o b e a v a i l -ab le f o r eve r ( Ba r t l e t t , 1986 ) .

L I N E A R G R A P H SI n F ig . 2 w e s e e a l in e a r g r a p h o f t w o p o p u l a t i o n s v s . t im e . B o t h p o p u -

l a t i o n s h a v e t h e v a l u e N = 1 0 0 p e o p l e a t t h e t i m e t = 0 . T h e lo w e r c u r v eis t h e c u r v e o f l in e a r g r o w t h i n w h i c h t h e p o p u l a t i o n in c re a s e s b y 1 0 p e o -p l e e v e r y y e a r . T h e e q u a t i o n o f t h i s l i n e i s

P = P o S t


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wh ere S is the s lope, w hic h in th is case is S = 10 peop le per year , and t isthe t ime in years.

P = 100 + 10 tThe uppe r cu rve i s t he curve o f s teady g row th in wh ich the popu la t ionstarts at 100 peo ple and increases at a rate of 10 per year.

P = 100 e (1 t)The s lope of the curves is the ra te o f change of the populat ion which

in i t ia l ly is the same for both pop ulat ions . For the low er cu rve, the s lope isconstant and has the va lue of 10 people per year a t a l l t imes. The uppercurve has the same in i t ia l s lope as the low er curve, but the u ppe r curvegets steeper as t im e goes on. By di f feren t iat ing Eq.3, we can f ind that thes lope of the upper curve is ,8 ) dN / dt = K No e (k t) = k N in peo ple p er yea r

Because i t dea ls w i th d i fferent ia ls , Eq.8 is va l id on ly for times m uchshor te r t han the doub l ing t ime.

The tw o curves of F ig .2 are com pare d in Table 3 . We need these tw ocurves and the data o f Table 3 to exp la in an impor tant puzz le .

TA B LE 3Com par ison of L inea r Gro wth and Steady Exponent ia l ) Grow th

L inear Growth Steady GrowthP opu la t i on Ra t eof change Pop ulat ion Rate of change

At end o f S ize o f popu la t ion S ize o f popu la t ionthe yea r People People per yea r People P/Y at end of yr .

Zero 100 10.0 100 10.01 110 10.0 111 11.12 120 10.0 122 12.23 130 10.0 135 13.54 140 10.0 149 14.95 150 10.0 165 16.5


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A P U Z Z L EW h e n t h e p o p u l a t i o n o f 1 0 0 p e o p l e g ro w s s t e a d i l y a t t h e ra te o f 1 0 %p e r y e a r , T a b l e 3 s h o w s t h a t t h e i n c r e a s e o f p o p u l a t i o n i n t h e fi rs t y e a r i s

n o t 1 0 p e o p l e , b u t i s 1 1 o r 1 1 % o f t h e i n i t i a l p o p u l a t i o n . H o w c a n a 1 0 %a n n u a l g r o w t h ra te g iv e a n a n n u a l i n cr e a s e o f 1 1 % ?The answ er is that a t the en d of one year, an ann ua l ra te o f 10

compounded con t inuous ly g ives a y ie ld wh ich i s t he same as an annua lrate o f 11 com po un de d once a year. T h i s e x p l a i n s b a n k a d s f o r s a v i n g sa c c o u n t s t h a t g i v e n u m b e r s s u c h a s t h e se : R a t e , 7 . 8 5 % ; Y i e l d , 8 . 1 7 % .T h e a d s t e l l u s t h a t $ 1 l e ft i n t h e b a n k f o r o n e y e a r a t a r a te o f 7 . 8 5 %c o m p o u n d e d c o n t i n u o u s l y ( g r o w i n g e x p o n e n t ia l l y ) w i l l , a t t h e e n d o f o n ey e a r , h a v e t h e v a l u e o f $ 1 . 0 8 1 7 b e c a u s e ,

e ( 7 8 5 x 1) = 1 . 0 8 1 7 . . .T h i s is t h e s a m e a n n u a l y i e l d a s o n e w o u l d h a v e a t a r at e o f 8 . 1 7 % c o m -p o u n d e d o n c e a y e a r .

L e t u s e x t e n d t h i s to a sk , w h a t is t h e d o u b l i n g t i m e i f o n e h a d s t e a d yg r o w t h a t t h e r a te o f 6 9 . 3 % p e r ye a r? E q . 6 s ug g es ts t h a t T 2 w o u l d b e o n ey e a r . H o w c a n a ra te o f 6 9 . 3 % p e r y e a r g i v e a n in c r e a se o f 1 0 0 % i n o n ey ea r? T h i s is a m o r e d r a m a t i c e x a m p l e o f t h e p u z z l e t h a t w e j u s t e x a m i n e da n d i t h a s t h e s a m e e x p l a n a t i o n . Steady growth at an annual ra te of 69.3g ives an an nua l y ie ld o f 100 .P e o p l e s o m e t i m e s m i s t a k e n l y s u g g e st t h a t E q . 6 is a p p r o x i m a t e , a n d isv a l i d o n l y f o r s m a l l ra te s o f g r o w t h . E q u a t i o n 6 is e x a c t a n d c o r r e c t fo r a l lr a t e s o f g r o w t h .

S E M I L O G A R I T H M I C G R A P H SO n a s e m i - lo g a r i t h m i c g r a p h , a s t r a ig h t l in e r e pr e se n ts s t e ad y g r o w t h .T h i s i s i m p o r t a n t i n r e c o g n i z i n g w h e t h e r o r n o t a se rie s o f d a t a p o i n t s

r e p r e s e n t s t e a d y g r o w t h . T o s e e t h i s p r o p e r t y , l e t u s t a k e th e n a t u r a l l o g -a r i t h m s o f b o t h s i d e s o f E q . 3 .In N = ( In No ) + k t

I f w e p l o t ( I n N ) v s . t w e h a v e a s t r a i g h t l i n e w h o s e s l o p e i s k a n d w h o s ein te r ce p t a t t im e t = 0 i s ( In No ).O n e u se s a s c i e n t i f ic c a l c u l a t o r t o l o o k u p t h e v a l u e s o f t h e lo g -


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T A B L E 4E a rly D a t a F r o m t h e U S C e n s u s

N a t . L o g L o g t o b a s e 1 0Y e a r P o p u l a t i o n I n l o g1790 3 ,929 ,214 15 .183 9. . . 6 .594 3. . .1800 5 ,308 ,48 3 15 .484 8. . . 6 .724 9. . .1810 7 ,239 ,881 15 .795 1. . . 6 .895 7. . .1820 9 ,638 ,453 16 .0812 . . . 6 .9840 . . .1830 12 ,860,702 16 .369 6. . . 7 .1092 . . .1840 17 ,063,353 16 .652 4. . . 7 .2320 . . .1850 23 ,191 ,876 16 .9593 . . . 7 .3653 . . .1860 31 ,443,321 17 .263 6. . . 7 .4975 . . .

a r i th m s . N a t u r a l lo g a r i th m s a re c a l l e d u p u s i n g t h e I n k e y , a n d l o g -a r i t h m s t o t h e b a se t e n a r e c a l l e d u p w i t h t h e l o g k e y . E i t h e r t y p e o fl o g a r i t h m c a n b e u s e d .EXAM PLE N O 6

T a b l e 4 g i v e s d a t a o n t h e e a r l y h i s t o r y o f t h e c e n s u s i n t h e U n i t e dS ta te s s h o w i n g , f o r e a c h d e c a d e , t h e p o p u l a t i o n , t h e n a t u r a l l o g a r i t h m o ft h e p o p u l a t i o n , a n d t h e l o g a r i t h m t o t h e b a s e t e n o f t h e p o p u l a t i o n . T w os e m i - lo g g r a p h s a re s h o w n i n F i g .3 . F i g .( 3 a ) s h o w s t h e n a t u r a l l o g a r i t h m sv s . t i m e f r o m T a b l e I V , a n d F i g .( 3 b ) s h o w s t h e l o g a r i th m s to t h e b a se t e nv s . t i m e . I n b o t h c a s es t h e p o i n t s fa l l r e m a r k a b l y c l o s e to s t r a i g h t li n e s . T h es t r a i g h t l i n e i s a v i s u a l c l u e t h a t th e p o p u l a t i o n o f t h e U n i t e d S ta te s g r e ws t e a d i l y (e x p o n e n t i a l l y ) i n t h e p e r i o d f r o m 1 7 9 0 t o 1 8 6 0 .

W e c a n c a l c u l a t e t h e a v e r a g e g r o w t h r a t e i n t h e 7 0 y e a r i n t e r v a l f r o m1 7 9 0 t o 1 8 6 0 .

k = (1 / 7 0 ) I n ( 3 1 , 4 4 3 , 3 2 1 / 3 , 9 2 9 , 2 1 4 )k = (1 / 7 0 ) 1 n ( 8 . 0 0 2 4 . . . )k = (1 / 7 0 ) 2 . 0 7 9 7 . . .k = 0 . 0 2 9 7 . . . o r R = 2 . 9 7 % p e r y e a r

F r o m E q . (6 ) , t h e d o u b l i n g t i m e , T 2 = 2 3 . 3 y e a r s.T h e n u m b e r s o f t h is e x a m p l e a r e p a r ti c u la r l y c o n v e n i e n t f o r a p o w e r s

o f t w o c a l c u l a t i o n , w h i c h o n e c a n d o i n o n e ' s h e a d. T h e 1 8 6 0 p o p u l a t i o n


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375

ALBERT A BARTLETT

FIGURE 3. L inear graphs of the logar i thms of the populat ion of theUn i ted States f rom 1790 to 1860. The upp er curve a) shows natura llogar ithms and the low er curve shows logar ithms to the b ase ten. In eachgraph the data points fal l very c lose to a straight l ine, which is v isualev idence that the p opu lat ion gro w th o f the U ni ted S tates in th is per iodwas s teady exponent ia l ) .

~ T o

I I i IL O I T O THE BASE e.OF U.S. POPULRTION

o - /

1 I ' 1

- 1 6 , 5 0 . ~

_ . 0

LOG TO TH E. BA SE IOO F U .5 . P O P U L A T I O N7 , 5

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is ve ry n ear l y e igh t t ime s the 1 790 po pu la t i on , (8 .0024 ) . E igh t is t h reedou b l i ngs , w h i c h to ok p lace i n 70 years . So T2 = 70 years / 3 = 23 years .

S E M I L O G A R I T H M I C G R A P H P APE R

T h e m o s t c o n v e n i e n t w a y t o m a k e a s e m i - lo g p l o t is t o u s e s e m i - lo ggraph paper . Th i s paper i s p r i n ted i n such a w ay tha t d is tances on thever t i ca l sca le a re p ropor t i ona l t o the l ogar i t hms o f t he quan t i t y be ing p lo t -ted, w h i le the ho r izon ta l sca le is the usual l ine ar sca le . Th is saves one theneed to l ook up l ogar i t hms .Impor tan t p roper t i es o f sem i - l og paper a re tha t a l l o f t he decades o fchan ge on th e ver t i ca l sca le are represented by the same d is tance, andthere i s no ze ro on the ve r t i ca l sca le . Thus the d i s tances on the ve r t i ca lsca le are the same for the nu mb ers f rom 1 to 10, as f rom 10 to 1 00, asf rom 100 to 1000 , e t c . Sem i - log g raph pa per can be purchased a t boo k -s to res tha t have eng ineer i ng supp l i es . I n buy ing sem i - l og paper , one mus ts p e c i fy h o w m a n y c y c le s o n e w a n t s . O n e - c y c l e p a p e r w i l l a c c o m o d a t eo n e d e c a d e o f d a t a ; t w o - c y c l e p a p e r w i l l h a n d l e t w o de ca d es o f d a t a , a sf o r e x a m p l e f ro m 1 0 0 0 t o 1 0 0 , 0 0 0 . F i v e - c y c l e p a p e r w i l l a c c o m m o d a t eda ta rang ing over f i ve decades .

In t h e e x a m p l e o f U n i t e d S t a t e s p o p u l a t io n f ro m 1 7 9 0 t o 1 8 6 0 , t h ep o p u l a t i o n n u m b e r s r an g e f ro m 3 . 9 m i l l io n t o 1 2 . 8 m i ll i o n . T h i s w i l l r e -q u i r e t w o - c y c l e p a p e r , o n e c y c l e f o r t h e ra n ge 1 t o 1 0 m i l li o n , a n d t h esecond cyc le fo r the range from 10 to 10 0 m i l l i on . Th i s g raph is sho wn i nF ig .4 . The da ta po in ts a re su r rounded by c i r c l es wh ich a re b racke ted byerror bars . In th is case the ver t i ca l er ror bars show an arb i t rary range ofp lus o r m inus 5% . The po in t i n sh ow ing these a rb i t ra ry e r ro r ba rs i s t osh ow th a t on a sem i - log g raph a f i xed f rac t i ona l e r ro r , such as 5% is rep re-sen ted by a cons tan t ve r t i ca l d i s tance , no m at te r wh ere on e is on thec u r v e .

D O U B L I N G T IM E S F R O M S E M I L O G G R A P H P A P E RThe g raph on sem i - l og paper a l l ows i n te rpo la t i ons and ex t rapo la t i ons .

F rom F ig .4 one can read tha t t he U .S . pop u la t i on reached 8 .0 m i l l ion a tthe s ta r t o f t he year 1814 and reached the popu la t i on o f 20 .0 m i l l i on abou tt h e d a t e 1 8 4 5 . 2 .T h e d o u b l i n g t i m e f o r t h e e a r l y g r o w t h o f t h e U . S . p o p u l a t i o n c a n b er ea d d i r e c t l y fr o m a g r a p h o n s e m i - lo g g r a p h p a p e r , w i t h o u t th e n e e d to d o


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3 7 7

ALBERT A BARTLETT

FIG URE 4 . S e m i -l o g g r a p h o f t h e p o p u l a t i o n o f t h e U n i t e d S t ate s f o r t h ep e r i o d 1 7 9 0 to 1 8 6 0 , p l o t te d o n t w o c y c l e s e m i -l o g g ra p h p a p e r . N o t i c et h a t t h e d i s ta n c e o n t h e v e r ti c a l a x is f r o m 1 m i l li o n t o 1 0 m i l li o n is t h es a m e as t h e d i s t a n c e f ro m 1 0 m i l l i o n t o 1 0 0 m i l li o n . A s in F i g .3 , t h ep o i n t s f a l l n i c e l y o n a s t r a i g h t l in e . T h e v e r t i c a l e r r o r b a r s i ll u s t r a t e t h es iz e o f a n h y p o t h e t ic a l u n c e r t a in t y o f p l u s o r m i n u s 5 % .- -~ I O 0 1 1 q I L L I O I ~ I I I I _

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T I M EI I II~ ~ O I ~ 0 I g S O


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T A B L E 5Times fo r a Succession of Dou bl ings

as Read From the Grap h o f F ig . 4 )P o p u l a t i o n , i n m i l l io n s D a t e , y e a rs

First Ser ies of Points5.0 1798.010.0 1821.620.0 1845.2Second Ser ies of Points4 .0 1790.58 .0 1814.016.0 1837.6

e x p o n e n t i a l a r i th m e t i c . W e n e e d to r e a d th e t im e s f o r a su c c e s s io n o f d o u -b l i n g s . I n F i g .4 , w e c a n r e a d t w o s ets o f p o i n t s a s s h o w n i n T a b l e 5 . I n t h ef i r s t s e t o f p o i n t s , t w o d o u b l i n g s w e r e o b s e r v e d i n t h e 4 7 . 2 y e a r s b e t w e e n1 7 9 8 . 0 a n d 1 8 4 5 . 2 . T h u s o n e d o u b l i n g t im e i s 4 7 . 2 / 2 ) = 2 3 . 6 y e a r s . I nt h e s e c o n d s e t o f p o i n t s , t w o d o u b l i n g s w e r e o b s e r v e d i n t h e 4 7 .1 y e a rsb e t w e e n 1 7 9 0 . 5 a n d 1 8 3 7 . 6 . T h u s o n e d o u b l i n g t i m e i s 4 7 .1 / 2 ) =2 3 . 5 5 y e a rs . T h e d i f f e r e n c e b e t w e e n t h es e t w o r e su lt s is i n d i c a t i v e o f t h ep r e c i s i o n o f t h i s g r a p h i c a l m e t h o d .

G R O W T H FA STE R O R S L O W E R T H A N E X P O N E N T IA LT a b l e 6 c o n t a i n s a s e t o f d a ta o n w o r l d p o p u l a t i o n o v e r a p e r io d o f

a b o u t 4 0 0 y e a rs W o r l d A l m a n a c , 1 9 9 2 ) . F i g u re 5 is a s e m i - lo g p l o t o ft h e se d a t a. F r o m a n e x a m i n a t i o n o f F i g. 5 w e c a n s ee t h a t f ro m 1 6 5 0 t oa b o u t 1 9 7 5 t h e l i n e is c u r v i n g m o r e s t e e p ly u p w a r d a s t i m e g o e s o n . T h i si n d i c a te s g r o w t h w h i c h is fas te r than expone nt ia l S o m e t i m e a f t e r 1 9 7 5a n d f o r t h e p r o j e c t i o n s t o t h e y e a r 2 0 2 5 , t h e c u r v e i s b e c o m i n g l e s s s t e e p ,w h i c h i n d i c a te s g r o w t h t h a t is s lower than exponent ia l

T h e t h i r d c o l u m n i n T a b l e 6 t a b u la t e s t h e a v e ra g e a n n u a l g r o w t h ra te sf o r e a c h o f t h e i n t e r v a l s in th e d a t a , a s c a l c u l a t e d f r o m E q . 4 , w h i l e t h ef o u r t h c o l u m n li st s t h e d a te s o f t h e m i d - p o i n t s o f e a c h o f th e i n t e r v a l s int h e d a t a .

F i g u re 6 s h o w s a p l o t o f th e t h i r d a n d f o u r t h c o l u m n s o f T a b l e 6 . H e r ew e s ee h o w t h e v a l u e o f R , t h e r a te o f g r o w t h o f w o r l d p o p u l a t i o n , h a s


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T A B L E 6An Est imate of the Recent H is tory o f W or ld Popula tion Gro w th 8 )wi th a Pro ject ion to the Year 2025

YearAverage rate

Popu la t ion pop . g row th M id da tein b i l l ion s per year o f per iod

1650 0.5500.276 17001750 0.725 0.483 18001850 1.175 0.617 19001900 1.60 0.744 19151930 2.00 1.23 19401950 2.56 1.78 1967.51975 4.00 2.27 1977.51980 4.48 1.83 19831986 5.00 1.60 19881990 5.33 1.27 1992.51995 5.68 1.52 1997.5

2000 6.13 1.15 2012.52025 8.18

chan ged, as ind icated by these data. T he y ind icate that the qua nt i t y Rinc reased to a m ax im um va lue o f app rox ima te ly 2 per year a round theyear 1975, and then began to dec l ine.

No c la im is made that these data are def in i t ive. They are used only asan i l lus t ra t ion. There are large uncer ta in t ies in the data on wor ld popula-t ion. I t w i l l be good news indeed i f fur ther s tud ies co nf i rm that R peakedand tha t wor ld popu la t ion g rowth has made the t rans i t i on f rom growth tha tis fas ter than expo ne nt ia l to s lowe r than expon ent ia l . H ow eve r , th is goodnews shou ld no t b l ind us t o t he fac t t ha t ze ro popu la t ion g row th w i l l no toc cu r unt i l R = 0.


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38


FIGURE 5. Semi-log graph of world population from 1650 withprojections to 2025. From 1650 to about 1975 the line is curvingupward, which represents growth faster than exponential. After about1985, the curve starts becoming less steep, which represents growthslower than exponential. The curve has an inflection point somewherebetween 1975 and 1985. The point at 1930 is shown with an error barof plus or minus 10%, solely to illustrate the size of an uncertaintyof this magnitude.I i I

~ I 0 D I L L I O N- 7.~ W O R L D POP U A'TtON-S

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F IGURE 6 . The rate o f g row th R o f w or ld po pu la t i on i n pe rcen t pe r yearas a func t i on o f t im e , as ca l cu la ted f rom the da ta show n i n F ig .5 . Thepo in ts sca t te r cons iderab l y , bu t t hey sugges t a peak g rowth ra te o f abou t2% per year was reached a roun d 1975 . The u ncer ta in t i es i n these ra tes o fg rowth cou ld be as l a rge as p lus o r m inus a ha l f pe rcen t .I I I I

PERCENT I NGP- .EASEPER. YEAR. IN-Z O WORLD P0PUI_I~TIONI . 0

~ 7 0 0 1

Q e

O,,O ~ l q O O 2 _ O O O1 1 8 o 0 1 I I I

The fac ts o f popula t ion growth are o f ten misrepresented.H u m a n p o p u l a t i o n s t a y e d a l m o s t c o n s t a n t a t n e a r l y 5 0 0 m i l l i o n f r o m

the year 0 to 15 00 . Then i t began r is i ng ex po ne n t i a l l y . I t dou b led be twe en1 8 5 0 a n d 1 9 5 0 , d o u b l e d a g a in b e tw e e n 1 9 5 0 a nd 1 9 9 0 . . . ( N e w s w e e k ,1992). The fac t t ha t t he second quo ted d ou b l i ng t ime is shor te r t han thef i r s t i nd i ca tes g rowth tha t i s fas ter than expone nt ia l I t w o u l d b e m o r e a c -c u r a te t o sa y t h a t t h e w o r l d p o p u l a t io n p r o b a b l y g r e w v e r y s l o w l y fr o m t h eda wn o f t im e up u n t i l a few hund red years ago. In the las t f ew cen tu r i esthe g row th rate has inc reased rap id l y wh i ch has resu lted i n po pu la t i ongrowth tha t has been fas te r t han exponen t i a l . The g rowth ra te may havep e ak ed a r o u n d 1 9 7 5 w i t h t h e c o n s e q u e n c e t h a t t h e g r o w t h s i n c e 1 9 7 5 h a sb e e n s l o w e r t h a n e x p o n e n t i a l .

Th e U ni ted States is c lose to ZPG w i th a fer t i l i t y ra te o f 2 .1 . . .(Popu la r Sc ience, 1992) . The U .S . Census f igu res fo r 198 0 and 199 0 sho wtha t the average g row th rate was 0 .9 33 % w h ic h trans la tes to a ne t inc reaseo f 2 .3 m i l l ion peop le each year. The a lleged und ercou n t , and i l lega l imm i -g ra t i on cou ld push th i s annua l i nc rease to 3 m i l l i on a year . The r i o t s i n ou rla rge c i ti es un de r l ine the fac t t ha t t he Un i ted S tates is no t no w ab le to takeprope r ca re o f t he p resen t U .S . p op u la t i on , and they emph as i ze the u r -gency o f reduc ing the U .S . popu la t i on g rowth ra te to ze ro and then tonega t ive va lues as rap id l y as poss ib le .


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S E R IE S O F E X M P L E SEXAMPLE NO 7

I n h i s la s t n i g h t a s h o s t o f t h e T o n i g h t s h o w , J o h n n y C a r s o n o b -s e rv e d t h a t w h e n t h e s h o w s t a rt ed o n O c t o b e r 1 , 1 9 6 2 t h e w o r l d p o p u l a -t i o n w a s 3 . 1 b i l l i o n , a n d a s h e w a s s i g n i n g o f f o n M a y 2 2 , 1 9 9 2 , h e s a idt h e w o r l d p o p u l a t i o n w a s 5 . 5 b i l l i o n . W h a t a v e ra g e g r o w t h ra te o f w o r l dp o p u l a t i o n is i n d i c a t e d b y t h e s e d a ta ?T h e s h o w r a n f o r 2 9 . 6 4 y e a rs , so th e v a l u e o f k is

k = (1 / 2 9 . 6 4 ) I n ( 5 . 5 / 3 . 1 ) = 0 . 0 1 9 3 . . . o r R = 1 . 9 3 % p e r y e a rC a r s o n ' s f ig u r e s i n d i c a t e a g r o w t h r at e s o m e w h a t h i g h e r t h a n t h e 1 . 7 % t h a tis q u o t e d f o r 1 9 9 2 .

W h a t w a s t h e w o r l d p o p u l a t i o n i n cr e a s e i n p e o p l e p e r d a y a t t h e s ta r ta n d e n d o f th e r u n o f J o h n n y C a r s o n 's s h o w a s i n d i c a t e d b y t h e d a t a C a r -s o n c it e d ? F o r t h is , w e n e e d t o k n o w t h e v a l u e o f k i n u n i ts o f per d a y .

k = 0 . 0 1 9 3 / 3 6 5 = 0 . 0 0 0 0 5 3 0 . . . p er d a yF r o m E q . 8 , t h e r a t e o f c h a n g e o f a p o p u l a t i o n is k N . T h i s g iv e s u s t h ef o l l o w i n g r a t e s o f c h a n g e

O c t , 1 9 6 2 ; 0 . 0 0 0 0 5 3 0 x 3 .1 x 1 09 = 1 . 6 4 x 1 05 p e o p l e p e r d a yM a y , 1 9 9 2 ; 0 . 0 0 0 0 5 3 0 x 5 . 5 x 1 09 = 2 .9 1 x 1 05 p e o p l e p e r d a y

I f w e w a n t t o k n o w t h e h o u r l y r a te o f c h a n g e , w e m u s t h a v e t h e v a lu e o f ki n u n i t s o f per hourk = 0 . 0 0 0 0 5 3 0 / 2 4 = 2 .2 0 8 x 1 0 6 p e r h o u r

O c t . 1 9 6 2 ; 2 . 2 0 8 x 1 0 - 6 x 3 .1 x 1 0 9 = 6 . 8 4 x 1 0 3 p e o p l e p e r h o u rM a y , 1 9 9 2 ; 2 . 2 0 8 1 0 _ 6 x 5 . 5 x 1 0 9 = 1 2 .1 1 0 3 p e o p l e p e r h o u rT h e s e n u m b e r s s h o w t h e d r a m a t i c c h a n g e s th a t h a v e ta k e n p l a c e i n t h ep e r i o d t h a t o n e p e r s o n h o s t e d a p o p u l a r t e l e v i s i o n s h o w .EXAMPLE NO 8

I t is r e p o r te d t h a t N o r t h A m e r i c a n w a t e r f o w l p o p u l a t i o n s h a v e d e -c l i n e d 3 0 % s i n c e 1 9 6 9 , m o s t l y d u e to th e l os s o f w e t l a n d h a b i t a t ( C le a r -i n g H o u s e B u l l e t i n , 1 9 9 2 ) .


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W h a t is t h e a v e r a g e a n n u a l r a te o f lo s s w h i c h w o u l d r e d u c e t h e w a t e r -f o w l p o p u l a t i o n t o 7 0 % o f it s o r i g i n a l p o p u l a t i o n in 2 2 y e a r s ? F r o m E q . 4K = (1 / 2 2 ) In ( 0 . 7 0 ) = (1 / 2 2 ) ( - 0 . 3 5 6 . . .) = - 0 . 0 1 6 2 . . .

T h e n u m b e r s i n d i c a t e t h e a v e r a g e r a te o f h a b i t a t l o ss o v e r t h e 2 2 y e a r s w a s1 . 6 2 % p e r y e a r .

A loss ra te o f 1 .62 pe r year is so sm al l as to seem to be t r i v ia l . Yet i tadds up to a ve ry s ign i f i can t l oss i n a modes t number o f years .

I f t h is r a te o f lo ss c o n t i n u e d f o r 5 0 y e a rs f r o m 1 9 6 9 t o 2 0 1 9 , w h a tf r a c ti o n o f t h e w a t e r f o w l w o u l d r e m a i n ?(N / No) = e - - O . 0 1 6 2 x 5 0 ) ~ e i - o . 8 1 1 = 0 . 4 4 . . .

O n l y a b o u t 4 4 % o f t he p o p u l a ti o n w o u l d r e m a in .E X A MP L E N O . 9

I t is r e p o r t e d th a t W o r l d f o o d p r o d u c t i o n w i l l h a v e t o i n cr e a s e t h r e e -f o l d i n t h e n e x t 4 0 y e a r s t o m e e t t h e n e e d s o f a n e s t i m a t e d n i n e b i l l i o np e o p l e ( Ga s s e r & F r a l e y , 1 9 9 2 ) .

W h a t is t h e a v e ra g e a n n u a l r a te o f in c r e a s e o f f o o d p r o d u c t i o n n e e d e dt o m e e t t h is g o a l ? T r i p l i n g i m p l i e s t h a t (N / N o ) = 3 . T h u s , f r o m E q . 4

k = (1 / 4 0 ) In 3 = 0 . 0 2 7 5 o r R = 2 . 7 5 % p e r y e a rEXAMPLE NO. 10

T h e p o p u l a t io n o f f is h e r m e n in C o l o r a d o h as e x p e r i e n c e d a f o u r f o ldi n c r e a s e i n t h e l a s t 3 5 y e a r s ( E n g le , 1 9 9 2 ) .

W h a t is t h e a v e ra g e r a te o f g r o w t h o f f is h e r m a n i n C o l o r a d o in th i sp e r i o d ? F r o m E q . 4 ,

k = (1 / 3 5 ) In 4 = 0 . 0 3 9 6 o r R = 3 . 9 6 % p e r y e a rT 2 = 6 9 . 3 / 3 . 9 6 = 1 7 . 5 y e a rs

N o t i c e t h a t t h e n u m b e r s in th i s e x a m p l e a l l o w u s t o w o r k t h is i n o u r h e a d .T h e f o u r f o ld i n cr e a s e is e x a c t l y t w o d o u b l i n g s . T w o d o u b l in g s in 3 5y e a r s m e a n s o n e d o u b l i n g in h a l f o f th i s, o r 1 7 . 5 y e a r s . T h e r a te o f g r o w t hc a n b e f o u n d b y d i v i d i n g 7 0 / 1 7 . 5 = 4 % p e r y e a r.


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E X A M P L E N O . 11I n 1 9 8 3 t h e s k i a r e a o f V a i l , C o l o r a d o c e l e b r a t e d i t s 2 0 t h a n n i v e r s a r y ,

a n d f o r a s h o r t p e r i o d a l l - d a y l i ft t i c k e t s w e r e s o l d a t t h e 1 9 6 3 p r i c e o f $ 5i n s t e a d o f a t t h e 1 9 8 3 p r i c e o f $ 2 0 . C a l c u l a t e t h e a v e r a g e r a t e o f i n f l a t i o no f the cos t o f t hese sk i l i f t t i c ke ts , and p r ed i c t t he cos t o f s k i l i f t t i c ke ts a tV a i l i f t h i s in f l a t i o n r at e c o n t i n u e s t o 1 9 9 3 a n d t o 2 0 0 3 .

T h e n u m b e r s i n t h is e x a m p l e a l l o w e a s y m e n t a l c a l c u l a t i o n . T h e co s to f l i f t t i c k e t s i n c r e a s e d b y a f a c t o r o f f o u r ( t w o d o u b in g s ) i n 2 0 y e a r s . T h ed o u b l i n g t i m e is t h e n 1 0 y e a r s . T h e n , fr o m E q . 6 , 7 0 / 1 0 = 7 % is t h ea v e r a g e a n n u a l r a te o f i n f l a t i o n o f l i f t t ic k e t s . I f t h i s ra t e c o n t i n u e s , l i ftt ic k e t s w i l l d o u b l e i n c o s t e v e r y d e c a d e ; t h e c o s t i n 1 9 9 3 w i l l b e $ 4 0 a n di n 2 0 0 3 i t w i l l b e $ 8 0 . . . T h e 1 9 9 2 c o s t o f l i f t t i c k e t s m a y a l r e a d y h a v ee x c e e d e d $ 4 0 .EXAMPLE NO 12

" T h e p o p u l a t i o n o f t h e t h r e e f o r m e r F r e n c h c o l o n i e s o f N o r t h A f r i c a ,T u n i s i a , A l g e r i a , a n d M o r o c c o , h a s n e a r l y t r ip l e d o v e r t h e p a s t t h r e e d e -cades , . . . " ( Randa l , 1992 ) .

W h a t i s t h e a v e r a g e r a t e o f g r o w t h t h a t g i v e s t r i p l i n g i n 3 0 y e a r s ?F r o m E q . 4 ,k = (1 / 30 ) In ( 3) = 0 . 03 6 6 o r R = 3 .6 6% pe r yea r

EXAMPLE NO 13

I t w a s n o t e d r e c e n t l y th a t t h e r e a re 7 6 0 , 0 0 0 l a w y e r s i n th e U n i t e dS ta te s i n 1 9 9 2 , a n d t h i s n u m b e r i s i n c r e a s i n g a t a n a n n u a l r at e o f 3 . 6 4 % . I tw a s a s s u m e d t h a t t h e a n n u a l r a t e o f i n c r e a s e o f t h e p o p u l a t i o n o f t h eU n i t e d S ta te s is " 0 . 6 % (a t y p i c a l f i g u r e ) , " a n d i t w a s t h e n c a l c u l a t e d t h a t i ft h e s e ra te s c o n t i n u e d , l a w y e r s w o u l d b e h a l f o f th e U n i t e d S t ate s p o p u l a -t i o n b y t h e y e a r 2 1 8 8 ( S e l i g m a n , 1 9 9 2 ) .T h e a v e r a g e a n n u a l g r o w t h r ate o f t h e U n i t e d S ta te s p o p u l a t i o n i n th e1 9 8 0 s w a s 0 . 9 3 3 % , r a t h e r t h a n 0 . 6 % . L e t u s d o t h e c a l c u l a t i o n u s i n g0 . 9 3 3 % . S i n c e t h e n u m b e r o f l a w y e r s is f o r 1 9 9 2 , w e m u s t e s t im a t e th eU n i t e d S t ate s p o p u l a t i o n i n 1 9 9 2 .

N1992 = 24 8 .7 1 1 0 6 e 0 .00933 x 2= 2 5 3 . 4 0 1 06 p e o p l e


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I f w e a s s u m e t h a t t h e g r o w t h r a t e s d o n o t c h a n g e , t h e e q u a t i o n f o r t h ep o p u l a t i o n o f t h e U n i t e d S ta te s a f te r 1 9 9 2 isNp = 2 5 3 .4 0 x 10 e 0 .00933 t

a n d t h e e q u a t i o n f o r t h e n u m b e r o f l a w y e r s i n t h e U n i t e d S t a t e s i sN L = 7 . 6 0 x 1 0 s e 0 0 3 6 4 t

T h e q u e s t i o n a sk s, w h e n w i l l l a w y e r s be h a l f o f t h e p o p u l a t i o n , o r w h e n isNL = Np / 2?

7 .6 0 x 105 e 0.0364 t = (2 5 3 .4 0 x 106 / 2) e 0.00933 tT h i s e q u a t i o n m u s t b e s o l v e d f o r t.

e fo .o 3 64 o .o o 93 3 ~t = 2 5 3 . 4 0 x 1 0 6 / ( 2 x 7 . 6 0 x 1 0 s )e o .o271t = 1 .6 67 102

0 . 02 71 t = I n ( 1 . 6 67 x 102 )t = 1 8 9 y e a r s

Np = 25 3 . 40 x 1 0 6 e 0 .00933 x 189w h i c h is a p p r o x i m a t e ly 1 . 5 b i l li o n p e o p l e , h a l f o f w h o m w o u l d b e la w y e rs .

I n t h i s e x a m p l e , t h e p o p u l a t i o n o f l a w y e r s i s g r o w i n g f a s t e r t h a n t h eg e n e r a l p o p u l a t i o n , w i t h t h e d i f fe r e n c e s i n t h e g r o w t h ra te s b e i n g d e -s c r i b e d b y k = 0 . 0 2 7 1 p e r y e a r . T h i s k h a s a d o u b l i n g t i m e ( E q .6 ) o f a b o u t2 5 y e a r s . S o a f te r 2 5 y e a r s p a s t t h e y e a r 2 1 8 1 , t h e e n t i r e U n i t e d S ta te sp o p u l a t i o n w o u l d b e l a w y e r s

T h e s e c a l c u l a t i o n s a re a n i c e e x a m p l e o f redu ct io ad absurdum. I t w a sp r o p o s e d t h a t t h e g r o w t h r a t e s o f t h e p o p u l a t i o n o f p e o p l e a n d o f l a w y e r sw o u l d r e m a i n c o n s t a n t f o r a l o n g p e r i o d o f t im e . T h i s l ed t o th e c o n c l u s i o nt h a t t h e U n i t e d S ta te s p o p u l a t i o n w o u l d b e 5 0 % l a w y e r s in 1 8 9 y e a r s. T h ea b s u r d i t y o f t h e c o n c l u s i o n p r o v e s t h e a b s u r d i t y o f th e a s s u m p t io n t h a t t h eg r o w t h ra te s c a n r e m a i n c o n s t a n t f o r l o n g p e r i o d s o f ti m e . Th us the term"sustainable growth" is an oxymoron

In th a t y e a r t h e p o p u l a t i o n o f th e U n i t e d S t ate s w o u l d b e


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3 8 6P O P U L A T IO N A N D E N V I R O N M E N T

C O N C L U S I O NI t is h o p e d t h a t t h i s t u t o r i a l , w i t h its w o r k e d o u t n u m e r i c a l e x a m p l e s ,

w i l l a s sis t r e ad e rs i n g a i n i n g a b e t t e r u n d e r s t a n d i n g o f th e n a t u r e o f g r o w t h ,a n d a n i m p r o v e d f a c i l it y i n i n t e r p r e t in g d a t a o n p o p u l a t i o n g r o w t h .

REFERENCESBar tl e tt , A .A . (Dec . 1969 ). The h ighw ay exp los ion . Civ i l Engineer ing pp . 71 -72 .Ba r tl e tt , A .A . (1978 ). The fo rgo t ten fund am en ta l s o f the ene rgy c r is is . Amer ican Journal ofPhysics 46 (9 ) , 876 -888 .Ba r tl e tt , A .A . (1986 ). Sus ta ined ava i l a b i l i t y : A m anag em en t p rog ram fo r non ren ew ab le re -sources. Amer ican Journal of Phys ics 54 (5 ) , 398 -402 .Clear ing House Bul let in (March , 1992 ) . A t t r i bu ted to The Chr is t ian S cience Mo ni tor ( D e c . 2 3 ,1 9 9 1 ). W a s h i n g t o n , D . C . : C a r r y i n g C a p a c i t y N e t w o r k .Eng le , E . (May 22, 1992) . Boulder Dai l y Camera B o u l d e r : C o l o r a d o .Gasser , C .S. Fra ley, R .T. (June 1992) . Transgen ic crops. Scienti f ic American pp . 62 - 69 .Lapp, R.E. (1973). The logari thmic century. N e w Y o r k : P r e n t i c e - H a l l .News week (June 1 , 1992) . p .34 .Randa l , J .C . (Fe b. 1 , 1992) . Washington Post W a s h i n g t o n , D . C . , p . A 1 6 .Se l igman, D. (June 15, 1992) . Ask Mr . S ta t is t ics , Fortune p . 1 5 9 .W or ld A lm anac an d B ook o f Fact s 1992 (1991) . N ew Y o rk : Pha ros Books , p .822 .

PPEND I XT h i s a p p e n d i x c o n t a i n s t h e f o r m u l a f o r t h e a v e r a g e o f a q u a n t i t y t h a t i sg r o w i n g e x p o n e n t i a l l y , a n d t h e k e y s tr o ke s n e e d e d t o w o r k a f e w o f t h ee x a m p l e s o n a s c i e n t i f ic c a l c u l a t o r .

I t c a n b e s h o w n t h a t w h e n N g r o w s i n a c c o r d w i t h E q . 3 , t h e a v e r a g eva lue o f N i n the i n te r va l f r om t = 0 to t = T is

. TfNay = (1 / T) / e kt d t = (N o / k T) [e kT - 1]

oE X A M P L E N O . 1

K E Y D I S P L A Y R E A D SO n 02 4 8 . 6 1 2 4 8 . 6 1D i v i s i o n s ig n 2 4 8 . 6 12 2 6 . 5 5 2 2 6 . 5 5E q u a ls s ig n 1 . 0 9 7 3 7 . . .I n 0 . 0 9 2 9 1 9 7 . . .


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D i v i s i o n s ig n 0 . 0 9 2 9 1 9 7 . . .10 10E q u als s ig n 0 . 0 0 9 2 9 1 9 7 . . .

Th i s l as t num be r is t he va lue o f k tha t we a re seek ing .E X A M P L E N O . 2

KEY DISPLAY READ SO n 02 2yX 22 . 1 4 2 . 1 4Equa ls s i gn 4 .4 07 . . .

EXAMPLE N O . 3 To Ca lcu la te 229 / 100KEY DISPLAY READ SO n 02 2yX 22 9 2 9E q ua ls s ig n 5 3 6 8 7 0 9 1 2D i v i s i o n s ig n 5 3 6 8 7 0 9 1 21 1E q ua ls s ig n 5 3 6 8 7 0 9 . 1 2

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