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

1kPdP

dt 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 log

cktP log

kPdPdt 1

Exponential Growth & Decay

kPdtdP

Growth and decay is proportional to population.

PdP

kt

1

cPk

log1

cPkt log

cktP logckteP

kPdPdt 1

Exponential Growth & Decay

kPdtdP

Growth and decay is proportional to population.

PdP

kt

1

cPk

log1

cPkt log

cktP logckteP

ckt eeP

kPdPdt 1

Exponential Growth & Decay

kPdtdP

Growth and decay is proportional to population.

PdP

kt

1

cPk

log1

cPkt log

cktP logckteP

ckt eeP

ktAeP

kPdPdt 1

Exponential Growth & Decay

kPdtdP

Growth and decay is proportional to population.

PdP

kt

1

cPk

log1

cPkt log

cktP logckteP

ckt eeP

ktAeP

P = population at time t

A = initial population

k = growth(or decay) constant

t = 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 hours21

3after 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 hours21

3after population theDetermine b)

teP

A

Pt

1.01000000

0000001

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 hours21

3after population theDetermine b)

teP

A

Pt

1.01000000

0000001

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 hours21

3after population theDetermine b)

teP

A

Pt

1.01000000

0000001

1000000,0when

bacteria 1419000 is therehours21

3after

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

kteP

A

Pt

1732

1732

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

kteP

A

Pt

1732

1732

1732,0when

ke

Pt1017321260 i.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

kteP

A

Pt

1732

1732

1732,0when

ke

Pt1017321260 i.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

kteP

A

Pt

1732

1732

1732,0when

ke

Pt1017321260 i.e.

1260,10when

1732126010 ke

17321260

log101

17321260

log10

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

kteP

A

Pt

1732

1732

1732,0when

ke

Pt1017321260 i.e.

1260,10when

1732126010 ke

17321260

log101

17321260

log10

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

kteP

A

Pt

1732

1732

1732,0when

ke

Pt1017321260 i.e.

1260,10when

1732126010 ke

17321260

log101

17321260

log10

k

k

0318165.0k

3%- is rategrowth

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

21

log1

21

log

21

kt

kt

ekt

b) In how many years will the population be half that in 1960?

kteP 1732866 ,866when

21

log1

21

log

21

kt

kt

ekt

786.21t

b) In how many years will the population be half that in 1960?

kteP 1732866 ,866when

21

log1

21

log

21

kt

kt

ekt

786.21thalved has population theyears 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

21

log1

21

log

21

kt

kt

ekt

786.21thalved has population theyears 22In