Chapter 5.7, Problem 55E

Population Growth. Before 1859, rabbits did not exist in Australia. That year, a settler released 24 rabbits into the wild. Without natural predators, the growth of the Australian rabbit population can be modeled by the uninhibited growth model $dP/dt=kP,$ where $P\left(t\right)$ is the population of rabbits t years after 1859. (Source: www.dpi.vic.gov.au/agriculture.)

a. When the rabbit population was established to be 8900, its rate of growth was about 2630 rabbits per year. Use this information to find k, and then find the particular solution of the differential equation.

b. Find the rabbit population in 1900 $\left(t=41\right)$ and the rate at which it was increasing in that year.

c. Without using a calculator, find $P^{\prime }\left(41\right)/P\left(41\right)$.