Exponential Growth & Decay
Exponential Growth & Decay
Exponential Growth & DecayGrowth and decay is proportional to population.
Exponential Growth & Decay
kPdtdP
Growth and decay is proportional to population.
Exponential Growth & Decay
kPdtdP
Growth and decay is proportional to population.
kPdPdt 1
Exponential Growth & Decay
kPdtdP
Growth and decay is proportional to population.
PdP
kt 1
kPdPdt 1
Exponential Growth & Decay
kPdtdP
Growth and decay is proportional to population.
PdP
kt 1
cPk
log1
kPdPdt 1
Exponential Growth & Decay
kPdtdP
Growth and decay is proportional to population.
PdP
kt 1
cPk
log1
cPkt log
kPdPdt 1
Exponential Growth & Decay
kPdtdP
Growth and decay is proportional to population.
PdP
kt 1
cPk
log1
cPkt logcktP log
kPdPdt 1
Exponential Growth & Decay
kPdtdP
Growth and decay is proportional to population.
PdP
kt 1
cPk
log1
cPkt logcktP log
ckteP
kPdPdt 1
Exponential Growth & Decay
kPdtdP
Growth and decay is proportional to population.
PdP
kt 1
cPk
log1
cPkt logcktP log
ckteP ckt eeP
kPdPdt 1
Exponential Growth & Decay
kPdtdP
Growth and decay is proportional to population.
PdP
kt 1
cPk
log1
cPkt logcktP log
ckteP ckt eeP
ktAeP
kPdPdt 1
Exponential Growth & Decay
kPdtdP
Growth and decay is proportional to population.
PdP
kt 1
cPk
log1
cPkt logcktP log
ckteP ckt eeP
ktAeP
P = population at time t
A = initial population
k = growth(or decay) constantt = time
kPdPdt 1
e.g.(i) The growth rate per hour of a population of bacteria is 10% of the population. The initial population is 1000000
e.g.(i) The growth rate per hour of a population of bacteria is 10% of the population. The initial population is 1000000
equation. aldifferenti theosolution t a is that Show a) 1.0 tAeP
e.g.(i) The growth rate per hour of a population of bacteria is 10% of the population. The initial population is 1000000
equation. aldifferenti theosolution t a is that Show a) 1.0 tAeP tAeP 1.0
e.g.(i) The growth rate per hour of a population of bacteria is 10% of the population. The initial population is 1000000
equation. aldifferenti theosolution t a is that Show a) 1.0 tAeP tAeP 1.0
tAedtdP 1.01.0
e.g.(i) The growth rate per hour of a population of bacteria is 10% of the population. The initial population is 1000000
equation. aldifferenti theosolution t a is that Show a) 1.0 tAeP tAeP 1.0
tAedtdP 1.01.0
P1.0
e.g.(i) The growth rate per hour of a population of bacteria is 10% of the population. The initial population is 1000000
equation. aldifferenti theosolution t a is that Show a) 1.0 tAeP tAeP 1.0
figures.t significan 4 correct to hours213after population theDetermine b)
tAedtdP 1.01.0
P1.0
e.g.(i) The growth rate per hour of a population of bacteria is 10% of the population. The initial population is 1000000
equation. aldifferenti theosolution t a is that Show a) 1.0 tAeP tAeP 1.0
figures.t significan 4 correct to hours213after population theDetermine b)
tePA
Pt
1.010000000000001
1000000,0when
tAedtdP 1.01.0
P1.0
e.g.(i) The growth rate per hour of a population of bacteria is 10% of the population. The initial population is 1000000
equation. aldifferenti theosolution t a is that Show a) 1.0 tAeP tAeP 1.0
figures.t significan 4 correct to hours213after population theDetermine b)
tePA
Pt
1.010000000000001
1000000,0when
tAedtdP 1.01.0
P1.0
0001941 1000000,5.3when 5.31.0
ePt
e.g.(i) The growth rate per hour of a population of bacteria is 10% of the population. The initial population is 1000000
equation. aldifferenti theosolution t a is that Show a) 1.0 tAeP tAeP 1.0
figures.t significan 4 correct to hours213after population theDetermine b)
tePA
Pt
1.010000000000001
1000000,0when
bacteria 1419000 is therehours213after
tAedtdP 1.01.0
P1.0
0001941 1000000,5.3when 5.31.0
ePt
(ii) On an island, the population in 1960 was 1732 and in 1970 it was 1260.
a) Find the annual growth rate to the nearest %, assuming it is proportional to population.
(ii) On an island, the population in 1960 was 1732 and in 1970 it was 1260.
a) Find the annual growth rate to the nearest %, assuming it is proportional to population.
kPdtdP
(ii) On an island, the population in 1960 was 1732 and in 1970 it was 1260.
a) Find the annual growth rate to the nearest %, assuming it is proportional to population.
kPdtdP
ktAeP
(ii) On an island, the population in 1960 was 1732 and in 1970 it was 1260.
a) Find the annual growth rate to the nearest %, assuming it is proportional to population.
kPdtdP
ktAeP
ktePA
Pt
17321732
1732,0when
(ii) On an island, the population in 1960 was 1732 and in 1970 it was 1260.
a) Find the annual growth rate to the nearest %, assuming it is proportional to population.
kPdtdP
ktAeP
ktePA
Pt
17321732
1732,0when
kePt
1017321260i.e.1260,10when
(ii) On an island, the population in 1960 was 1732 and in 1970 it was 1260.
a) Find the annual growth rate to the nearest %, assuming it is proportional to population.
kPdtdP
ktAeP
ktePA
Pt
17321732
1732,0when
kePt
1017321260i.e.1260,10when
1732126010 ke
(ii) On an island, the population in 1960 was 1732 and in 1970 it was 1260.
a) Find the annual growth rate to the nearest %, assuming it is proportional to population.
kPdtdP
ktAeP
ktePA
Pt
17321732
1732,0when
kePt
1017321260i.e.1260,10when
1732126010 ke
17321260log
101
17321260log10
k
k
(ii) On an island, the population in 1960 was 1732 and in 1970 it was 1260.
a) Find the annual growth rate to the nearest %, assuming it is proportional to population.
kPdtdP
ktAeP
ktePA
Pt
17321732
1732,0when
kePt
1017321260i.e.1260,10when
1732126010 ke
17321260log
101
17321260log10
k
k
0318165.0k
(ii) On an island, the population in 1960 was 1732 and in 1970 it was 1260.
a) Find the annual growth rate to the nearest %, assuming it is proportional to population.
kPdtdP
ktAeP
ktePA
Pt
17321732
1732,0when
kePt
1017321260i.e.1260,10when
1732126010 ke
17321260log
101
17321260log10
k
k
0318165.0k
3%-isrategrowth
b) In how many years will the population be half that in 1960?
b) In how many years will the population be half that in 1960?kteP 1732866 ,866when
b) In how many years will the population be half that in 1960?kteP 1732866 ,866when
21log1
21log
21
kt
kt
ekt
b) In how many years will the population be half that in 1960?kteP 1732866 ,866when
21log1
21log
21
kt
kt
ekt
786.21t
b) In how many years will the population be half that in 1960?kteP 1732866 ,866when
21log1
21log
21
kt
kt
ekt
786.21thalved haspopulationtheyears 22In
Exercise 7G; 2, 3, 7, 9, 11, 12
b) In how many years will the population be half that in 1960?kteP 1732866 ,866when
21log1
21log
21
kt
kt
ekt
786.21thalved haspopulationtheyears 22In