Fifty elk are introduced into a game preserve. At what rate is the population increasing when t = 2? After how many years is the population increasing most rapidly?

Question

It is estimated that their population will increase according to the model P(t)=250/(1+4e^-t/3), where t is measured in years.

Answer 1

The rate of increase of the elk population P(t) is given by the time derivative dP/dt = 1000/3 e^-t/3 / (1+4e^-t/3)².

Substituting t = 2, we find that the population increases by approximately 18.353 elks/year after 2 years.

To find the value of t when dP/dt is maximal, we find the time derivative of dP/dt and set it equal to zero. The result is d²P/dt² = 1000/9 e^-t/3 (4e^-t/3 - 1) / (1+4e^-t/3)³ = 0. Solving this equation, we have 4e^-t/3 - 1 = 0, which simplifies to e^-t/3 = 1/4. Taking ln() of both sides, we obtain -t/3 = -2 ln(2). Therefore, the maximal rate of population increase is after t = 6 ln(2), which is approximately 4.159 years.