Logistic Growth - AP Calculus AB Exam Question
Consider a population of rabbits in a closed ecosystem that exhibits logistic growth. The population size (P) at time t (in years) is given by the equation:
P(t) = 800 / (1 + 2e^(-0.5t))
Solution:
1. Determine the rate of population growth at time t = 3 years.
To find the rate of population growth, we need to take the derivative of the population function with respect to time (t).
Given, P(t) = 800 / (1 + 2e^(-0.5t))
Differentiating both sides with respect to t:
dP/dt = (d/dt) [800 / (1 + 2e^(-0.5t))]
Using the quotient rule, we have:
dP/dt = [(-800)(d/dt)(1 + 2e^(-0.5t)) - (1 + 2e^(-0.5t))(d/dt)(800)] / (1 + 2e^(-0.5t))^2
Simplifying further:
dP/dt = [(-800)(0 + 2e^(-0.5t)(-0.5)) - (1 + 2e^(-0.5t))(0)] / (1 + 2e^(-0.5t))^2
dP/dt = [800e^(-0.5t) - 800e^(-t)] / (1 + 2e^(-0.5t))^2
Now, substitute t = 3 years into the equation to find the rate of growth at that time:
dP/dt (t = 3) = [800e^(-0.5(3)) - 800e^(-3)] / (1 + 2e^(-0.5(3)))^2
dP/dt (t = 3) = [800e^(-1.5) - 800e^(-3)] / (1 + 2e^(-1.5))^2
Calculating the numerical value:
dP/dt (t = 3) ≈ 7.107 rabbits per year
Therefore, the rate of population growth at t = 3 years is approximately 7.107 rabbits per year.
2. Find the population size at time t = ∞.
To find the population size at t = ∞, we need to examine the behavior of the population function as t approaches infinity.
As t approaches infinity, the exponential term e^(-0.5t) approaches 0, making the denominator of the population function approach 1. Hence, at t = ∞, the population size is given by:
P(t = ∞) = 800
Therefore, the population size at t = ∞ is 800 rabbits.
3. Calculate the time it takes for the population size to reach 600 rabbits.
To find the time it takes for the population size to reach 600 rabbits, we can set the population function equal to 600 and solve for t.
Given, P(t) = 800 / (1 + 2e^(-0.5t))
Setting P(t) = 600:
600 = 800 / (1 + 2e^(-0.5t))
Simplifying:
1 + 2e^(-0.5t) = 4/3
2e^(-0.5t) = 1/3
e^(-0.5t) = 1/6
Taking natural logarithm (ln) on both sides:
-0.5t = ln(1/6)
t = -2ln(1/6) / 0.5
Calculating the numerical value:
t ≈ 4.158 years
Therefore, it takes approximately 4.158 years for the population size to reach 600 rabbits.