Suppose That a Population, P, Grows at a Rate Given $\frac { d P } { d t } = b P e ^ { - k t }$

Essay

Suppose that a population, P, grows at a rate given by the equation $\frac { d P } { d t } = b P e ^ { - k t }$ , where P is the population (in thousands) at time t (in hours), and b and k are positive constants.(a) Find the solution to the differential equation when b = 0.04, k = 0.01 and P (0) = 1.(b) Find P (10), P (100), and P (1000).(c) After how many hours does the population reach 2 thousand? 30 thousand? 54 thousand?
(d) As time t increases without bound, what happens to the population?
(e) Sketch the graph of the solution of the differential equation.