abreviatsjw

2022-01-03

The population of a certain country is known to increase at a rate proportional to the number of people presently living in the country. If after two years the population has doubled, and after three years the population is 20,000. Estimate the initial population of the nation and find an expression for the country's approximate population at any time.

sirpsta3u

Beginner2022-01-04Added 42 answers

Given that the number of people living there now determines how quickly the population of the country grows,

$\frac{dP}{dt}=kP$, where P is the population

k is the proportionality constant

$\int \frac{1}{p}dP=\int kdt$

$\mathrm{ln}P=kt+C$

Let $P}_{0$ be the population initially

$t=0\Rightarrow P={P}_{0}$

$P={P}_{0}{e}^{kt}$

Given that at 2 years the population is dobuled

$2={e}^{2k}$

$k=\mathrm{ln}\left(\sqrt{2}\right)$

Given that at 3 years the population is 20000

$20000={P}_{i}{e}^{3\mathrm{ln}\sqrt{2}}$

Therefore the equation is $P={P}_{0}{e}^{\frac{\mathrm{ln}\left(2\right)}{2}t}$

Paul Mitchell

Beginner2022-01-05Added 40 answers

Rate of increase of population is proportional to Number of people living presently

After 2 years, Population has doubled

After 3 years, Population$=20000$

Let P be population in the country.

Thus, according to the question:

$\frac{dP}{dt}\mathrm{\infty}P$

$\frac{dP}{dt}=kP$ , where k is any constant.

$\frac{dP}{P}=k.dt$

$\int \frac{dP}{P}=\int k.dt$

$\mathrm{ln}P=k.t+C$

When$t=0$ years, $P={P}_{0}$ , where $P}_{0$ is the initial population.

Thus,${\mathrm{ln}P}_{0}=0+C$

$C={\mathrm{ln}P}_{0}$

$\mathrm{ln}P=kt+{\mathrm{ln}P}_{0}$

$\mathrm{ln}P-{\mathrm{ln}P}_{0}=kt$

$\mathrm{ln}\left(\frac{P}{{P}_{0}}\right)=kt$

When$t=2$ years, $P=2{P}_{0}$

$\mathrm{ln}\left(\frac{2{P}_{0}}{{P}_{0}}\right)=kx2$

$k=\frac{\mathrm{ln}2}{2}$

$\mathrm{ln}\left(\frac{P}{{P}_{0}}\right)=\frac{\mathrm{ln}2}{2}.t$

When$t=3$ years, $P=20000$

$\mathrm{ln}\left(\frac{20000}{{P}_{0}}\right)=\frac{\mathrm{ln}2}{2}x3$

$\mathrm{ln}\left(\frac{20000}{{P}_{0}}\right)=1.0397$

$\frac{20000}{{P}_{0}}={e}^{1.0397}$

${P}_{0}=\frac{20000}{{e}^{1.0397}}=7071.2$

${P}_{0}=7072$

So, initial population,${P}_{0}=7072$

After 2 years, Population has doubled

After 3 years, Population

Let P be population in the country.

Thus, according to the question:

When

Thus,

When

When

So, initial population,

karton

Expert2022-01-09Added 613 answers

Let make N represent the population of the country at any given time t and

where k is the constant of proportionality.

For

Therefore,

For

Substitute these values into (1)

Therefore (1) becomes

For

Substitute these values into (2)

Therefore the number of people initially living in the country is 7062

madeleinejames20

Skilled2023-06-14Added 165 answers

Eliza Beth13

Skilled2023-06-14Added 130 answers

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