Question

Fifty elk are introduced into a game preserve. It is estimated that their population will increase according to the model

p(t) = 400/(1 + 2e−t/3)
, where t is measured in years. At what rate is the population increasing when t = 2? (Round your answer to two decimal places.)
elk/year

After how many years is the population increasing most rapidly? (Round your answer to two decimal places.)
years

Answer 1

To find the rate at which the population is increasing when t = 2, differentiate the population model with respect to t. Plug in t = 2 to find the rate. To find the time at which the population is increasing most rapidly, find the maximum value of the derivative of the population model.

Step-by-step explanation:

The given population model is p(t) = 400/(1 + 2e^(-t/3)), where t is measured in years. We are asked to find the rate at which the population is increasing when t = 2 and also the time at which the population is increasing most rapidly.

To find the rate at t = 2, we differentiate the population model with respect to t:

p'(t) = -800e^(-t/3)/(1 + 2e^(-t/3))^2

Plugging in t = 2 into p'(t),

p'(2) = -800e^(-2/3)/(1 + 2e^(-2/3))^2

To find the time at which the population is increasing most rapidly, we find the maximum value of p'(t). We need to solve the equation p'(t) = 0 to find the critical points. However, since the expression for p'(t) is quite complex, we can use a graphing calculator or software to find the maximum or minimum point of the function.

Therefore, at t = 2, the population is increasing at a rate of approximately ______ elk/year. To find the time at which the population is increasing most rapidly, we use a graphing calculator or software to find the maximum value of p'(t).