Mathematics: In one model of the changing population P(t) of a community, it is assumed that dP dt = dB dt − dD dt , where dB/dt and dD/dt are the birth and death rates, respectively. (a) Solve for P(t) if dB/dt = k1P and dD/dt = k2P. (Assume P(0) = P0.)

In one model of the changing population P(t) of a

Ответ:

nicpinela1234

19.01.2022

$P(t) = P_{0}exp^{(k_{1}-k_{2})t\\$

Step-by-step explanation:

$\frac{dP}{dt}= \frac{dB}{dt}-\frac{dD}{dt}\\where \frac{dB}{dt}=k_{1}P ; and \frac{dD}{dt}=k_{2}P\\ Therefore:\\\frac{dP}{dt}=k_{1}P-k_{2}P\\\frac{dP}{dt}=(k_{1}-k_{2})P\\$

This leads to:

$\frac{dP}{P}=(k_{1}-k_{2})dt\\$

Taking the integral of both sides

$\int {\frac{dP}{P}=\int(k_{1}-k_{2})dt} \\ln P = (k_{1}-k_{2})t+C, C=constant of integration$ ......(i)

$P(t) = Cexp^{(k_{1}-k_{2})t\\$

When t=0, P(t)= $P_{0}$

From (i), $P_{0} = C\\$

Therefore: $P(t) = P_{0}exp^{(k_{1}-k_{2})t\\$

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Ответ:

nathansnyder200

28.10.2022

Mathematics

The area of the sector is 20π.

Step-by-step explanation:

The area of the circle = 100πThe formula for area of the circle = πr²The central angle of the sector = (2/5)π radians

The central angle should be converted from radians to degrees.

To convert radian to degree :

1 rad × 180/π gives the degree value.

Therefore, (2/5)π × 180/π = (2/5) × 180

The π gets cancelled out.

⇒ 2× 36

⇒ 72°

To find the area of the sector :

The formula to calculate area of the sector = (central angle/360) × πr²

⇒ (72/360) × 100π

⇒ (1/5) × 100π

⇒ 20π

The area of the sector is 20π.

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In one model of the changing population P(t) of a community, it is assumed that dP dt = dB dt − dD dt , where dB/dt and dD/dt are the birth and death rates, respectively. (a) Solve for P(t) if dB/dt = k1P and dD/dt = k2P. (Assume P(0) = P0.)

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