Assuming P≥0, suppose that a population develops according to the logistic equation
dP/dt=0.03P−0.00015P^2
where tt is measured in weeks. Answer the following questions.
1. What is the carrying capacity?
I tried solving the differential equation and got 600 but it says it's wrong what am I doing wrong??
Carrying Capacity:
1
2. What is the value of kk?
Answer: k=k=
1
3. For what values of PP is the population increasing?
Answer (in interval notation):
4. For what values of PP is the population decreasing?
Answer (in interval notation):
dp/dt=0.03p−0.00015p^2
This is a Bernoulli equation, with solution
200 e^0.03t / (e^c+e^0.03t)
No idea what k is supposed to be, or the carrying capacity. You will need some more info to determine c.
Since dp/dt = 0.00015p(200-p)
its roots are at p=0 and p=200
So, p is growing until it reaches 200, then starts decreasing. The problem is, p never reaches 200. So, p is always increasing. Read up on logistic growth.
1. What is the carrying capacity?
Well, I hate to break it to you, but the carrying capacity is actually not 600. It seems like your calculations went a little off. Let me lend you a helping hand and solve it for you:
0.03P - 0.00015P^2 = 0
Simplifying it further, we get:
0.00015P^2 - 0.03P = 0
Dividing by 0.00015, we have:
P^2 - 200P = 0
Factoring out P, we get:
P(P - 200) = 0
So, P can be either 0 or 200. But since we're assuming P≥0, the carrying capacity in this case is 200.
2. What is the value of k?
Ah, k, my dear friend. In this equation, there doesn't seem to be any k involved. So, I'm afraid the value of k is nonexistent. It's like trying to search for meaning in a knock-knock joke – it just doesn't make sense.
3. For what values of P is the population increasing?
To find the values of P where the population is increasing, we need to examine the derivative of the population function. Taking the derivative of dP/dt = 0.03P - 0.00015P^2, we get:
dP/dt = 0.03 - 0.0003P
Setting this derivative greater than zero to find where the population is increasing:
0.03 - 0.0003P > 0
Simplifying:
0.03 > 0.0003P
Dividing both sides by 0.0003:
100 > P
So, the population is increasing when P is less than 100. In interval notation, this can be written as (negative infinity, 100).
4. For what values of P is the population decreasing?
Similarly, we check the derivative dP/dt to find where the population is decreasing:
0.03 - 0.0003P < 0
Simplifying:
0.03 < 0.0003P
Dividing both sides by 0.0003:
100 < P
Therefore, the population is decreasing when P is greater than 100. In interval notation, this can be written as (100, positive infinity).
Hope this helps! And remember, math can be confusing, but never as confusing as why clowns wear such big shoes.
1. To find the carrying capacity, we need to solve the differential equation when dP/dt = 0. This occurs when the population growth rate is zero.
0 = 0.03P - 0.00015P^2
Simplifying the equation, we have:
0 = P(0.03 - 0.00015P)
Setting each factor equal to zero, we get:
P = 0 (which represents the trivial solution)
0.03 - 0.00015P = 0
Solving for P, we have:
0.03 - 0.00015P = 0
0.00015P = 0.03
P = 0.03 / 0.00015
P ≈ 200
So, the carrying capacity is 200.
2. The value of k in the logistic equation is the maximum growth rate, which is given by the coefficient of P in the equation. In this case, k = 0.03.
3. To determine for what values of P the population is increasing, we need to look at the sign of the population growth rate, dP/dt.
dP/dt = 0.03P - 0.00015P^2
The population is increasing when dP/dt > 0.
0.03P - 0.00015P^2 > 0
To solve this inequality, we can factor out P:
P(0.03 - 0.00015P) > 0
The inequality is positive when both factors have the same sign. So, we have two cases:
Case 1: P > 0 and (0.03 - 0.00015P) > 0
Solving this inequality, we find: 0 < P < 200
Case 2: P < 0 and (0.03 - 0.00015P) < 0
Since the population cannot be negative, we ignore this case.
Therefore, the population is increasing for the interval (0, 200).
4. To determine for what values of P the population is decreasing, we again look at the sign of the population growth rate, dP/dt.
dP/dt = 0.03P - 0.00015P^2
The population is decreasing when dP/dt < 0.
0.03P - 0.00015P^2 < 0
To solve this inequality, we can factor out P:
P(0.03 - 0.00015P) < 0
Again, we have two cases:
Case 1: P > 0 and (0.03 - 0.00015P) < 0
Solving this inequality, we find: P > 200
Case 2: P < 0 and (0.03 - 0.00015P) > 0
Since the population cannot be negative, we ignore this case.
Therefore, the population is decreasing for the interval (200, ∞).
To find the carrying capacity of the population, we need to set the derivative equation equal to zero and solve for P. In this case, the equation is:
dP/dt = 0.03P - 0.00015P^2
Setting this equation to zero, we have:
0.03P - 0.00015P^2 = 0
Now, let's solve for P by factoring out P from the equation:
P(0.03 - 0.00015P) = 0
This equation is satisfied when either P = 0 or (0.03 - 0.00015P) = 0.
For P = 0, the population size would be 0, which doesn't make sense as the carrying capacity.
To find the non-zero value for P, we need to solve the equation (0.03 - 0.00015P) = 0:
0.03 - 0.00015P = 0
Simplifying the equation, we have:
-0.00015P = -0.03
Dividing both sides by -0.00015:
P = -0.03 / -0.00015
P = 200
Therefore, the carrying capacity of the population is 200.
Now, let's address the issue with your answer of 600. It seems that you made an error while solving the quadratic equation. Double-check the algebraic steps you took to solve the equation for P.
For the value of k in the logistic equation, it is not explicitly mentioned in the given equation. It is possible that k refers to the growth rate constant, which is usually denoted as "r" or "k" in the logistic equation. However, without further information, we cannot determine its value based solely on the given equation.
To determine whether the population is increasing or decreasing, we need to observe the sign of the derivative equation (dP/dt) for different values of P.
Since dP/dt = 0.03P - 0.00015P^2, the population will be increasing when dP/dt is positive (greater than zero) and decreasing when dP/dt is negative (less than zero).
To find the values of P where the population is increasing, we need to find the intervals where dP/dt > 0:
0.03P - 0.00015P^2 > 0
Simplifying the inequality, we have:
0.03P(1 - 0.000005P) > 0
To determine the sign of the inequality, we need to consider the signs of each factor:
0.03P > 0 (P > 0)
and
1 - 0.000005P > 0
To solve the second inequality, we isolate P:
1 > 0.000005P
P > 1 / 0.000005
P > 200,000
Therefore, the population will be increasing for P values greater than 200,000.
To find the values of P where the population is decreasing, we need to find the intervals where dP/dt < 0:
0.03P - 0.00015P^2 < 0
Simplifying the inequality, we have:
0.03P(1 - 0.000005P) < 0
To determine the sign of the inequality, we need to consider the signs of each factor:
0.03P < 0 (P < 0)
and
1 - 0.000005P < 0
Solving the second inequality, we isolate P:
1 < 0.000005P
P < 1 / 0.000005
P < 200,000
Therefore, the population will be decreasing for P values less than 200,000.
In summary:
1. The carrying capacity is 200.
2. The value of k cannot be determined with the given information.
3. The population is increasing for P > 200,000.
4. The population is decreasing for P < 200,000.