A wildlife refuge introduces a population of deer into a large
MATHEMATICS · Multi-part · AP · Calculus BC · Differential Equations · Differential Equations
Problem
A wildlife refuge introduces a population of deer into a large, fenced preserve. Biologists model the deer population D(t), measured in animals, t years after introduction by the differential equation
dtdD=kD(6000−D),D(0)=400,
where k is a positive constant.
(a) Write an explicit general solution for D(t) in terms of the constant k.
(b) After 3 years, surveys count 800 deer. Use this information to determine the value of k, correct to four decimal places.
(c) Using the value of k obtained in part (b), predict the deer population after 10 years. Give your answer to the nearest whole number.
(d) Determine t→∞limD(t) and interpret the result in the context of the preserve.
(e) Compute the time (in years) at which the population is increasing most rapidly, and state the corresponding deer population at that time.