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Elina [12.6K]
3 years ago
14

Suppose a certain population satisfies the logistic equation given by dP

Mathematics
1 answer:
Ksenya-84 [330]3 years ago
8 0

Answer:

The population when t = 3 is 10.

Step-by-step explanation:

Suppose a certain population satisfies the logistic equation given by

\frac{dP}{dt}=10P-P^2

with P(0)=1. We need to find the population when t=3.

Using variable separable method we get

\frac{dP}{10P-P^2}=dt

Integrate both sides.

\int \frac{dP}{10P-P^2}=\int dt             .... (1)

Using partial fraction

\frac{1}{P(10-P)}=\frac{A}{P}+\frac{B}{(10-P)}

A=\frac{1}{10},B=\frac{1}{10}

Using these values the equation (1) can be written as

\int (\frac{1}{10P}+\frac{1}{10(10-P)})dP=\int dt

\int \frac{dP}{10P}+\int \frac{dP}{10(10-P)}=\int dt

On simplification we get

\frac{1}{10}\ln P-\frac{1}{10}\ln (10-P)=t+C

\frac{1}{10}(\ln \frac{P}{10-P})=t+C

We have P(0)=1

Substitute t=0 and P=1 in above equation.

\frac{1}{10}(\ln \frac{1}{10-1})=0+C

\frac{1}{10}(\ln \frac{1}{9})=C

The required equation is

\frac{1}{10}(\ln \frac{P}{10-P})=t+\frac{1}{10}(\ln \frac{1}{9})

Multiply both sides by 10.

\ln \frac{P}{10-P}=10t+\ln \frac{1}{9}

e^{\ln \frac{P}{10-P}}=e^{10t+\ln \frac{1}{9}}

\frac{P}{10-P}=\frac{1}{9}e^{10t}

Reciprocal it

\dfrac{10-P}{P}=9e^{-10t}

P(t)=\dfrac{10}{1+9e^{-10t}}

The population when t = 3 is

P(3)=\dfrac{10}{1+9e^{-10\cdot 3}}

Using calculator,

P=9.999\approx 10

Therefore, the population when t = 3 is 10.

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