A wildlife reserve introduces 80 rabbits into a protected meadow at time t=0 months. Biologists believe that, in the absence of predators, the rabbit population grows so that its instantaneous rate of change is proportional to its current size. They also observe that the population doubles every 7 months. Let R(t) be the rabbit population t months after introduction.
(1) Construct a differential equation that models the growth of the rabbit population.
(2) Find the value of the proportionality constant k using the doubling-time information.
(3) Determine the projected population after 18 months. Round your answer to the nearest whole rabbit.
(4) Calculate the instantaneous rate of change of the population at t=18 months and interpret this quantity in context, including correct units.