Respuesta :
Answer:
[tex]\frac{dP}{dt} = 2.4P(1 - \frac{P}{1200})[/tex]
Step-by-step explanation:
The logistic differential equation is as follows:
[tex]\frac{dP}{dt} = rP(1 - \frac{P}{K})[/tex]
In this problem, we have that:
[tex]K = 1200[/tex], which is the carring capacity of the population, that is, the maximum number of people allowed on the beach.
At 10 A.M., the number of people on the beach is 200 and is increasing at the rate of 400 per hour.
This means that [tex]\frac{dP}{dt} = 400[/tex] when [tex]P = 200[/tex]. With this, we can find r, that is, the growth rate,
So
[tex]\frac{dP}{dt} = rP(1 - \frac{P}{K})[/tex]
[tex]400 = 200r(1 - \frac{200}{1200})[/tex]
[tex]166.67r = 400[/tex]
[tex]r = 2.4[/tex]
So the differential equation is:
[tex]\frac{dP}{dt} = rP(1 - \frac{P}{K})[/tex]
[tex]\frac{dP}{dt} = 2.4P(1 - \frac{P}{1200})[/tex]
The differential equation which can describe this situation is [tex]\rm \dfrac{dP}{dt}=2.4P(1-\dfrac{P}{1200})[/tex] and this can be determined by using the given data and arithmetic operations.
Given :
- The rate of change (dP/dt), of the number of people on an ocean beach is modeled by a logistic differential equation.
- The maximum number of people allowed on the beach is 1200.
- At 10 A.M., the number of people on the beach is 200 and is increasing at the rate of 400 per hour.
The logistic differential equation is given by:
[tex]\rm \dfrac{dP}{dt}=rP(1-\dfrac{P}{K})[/tex] ----- (1)
Given that at 10 a.m. :
K = 1200
[tex]\rm \dfrac{dP}{dt} = 400[/tex]
P = 200
Now, put the values of K, P and dP/dt in the equation (1) to find the value of 'r'.
[tex]400=200r(1-\dfrac{200}{1200})[/tex]
[tex]2 = r\times \dfrac{1000}{1200}[/tex]
[tex]r =\dfrac{2400}{1000}[/tex]
r = 2.4
Now, put the value of r and K in the equation (1).
[tex]\rm \dfrac{dP}{dt}=2.4P(1-\dfrac{P}{1200})[/tex]
The differential equation which can describe this situation is [tex]\rm \dfrac{dP}{dt}=2.4P(1-\dfrac{P}{1200})[/tex] .
For more information, refer to the link given below:
https://brainly.com/question/16240546
