9. The rate of change of the population of a town in Pennsylvania at any time t is proportional to the population at that time. Four years ago, the population was 25,000. Now, the population is 36,000. Calculate what the population will be six years from now.
A. 43,200 |
B. 52,500 |
C. 62,208 |
D. 77,760 |
E. 89,580 |
Solution: The answer is C
Since the rate of change of the population is proportional to the current population, we have the following differential equation,
Where, k is the proportionality constant. The above differential equation can be solved, by integrating both sides after the variables and their corresponding differentials are separated, as follows:
Where C, is the constant of integration. Let P(0) = 25,000 (the population four years ago), so that by substituting into equation (1) above, C = 25,000 and P(t) = 25,000e^{kt}. Since the current population is 36,000, let P(4) = 36,000 to solve for k as follows,
So, to calculate the population six years from now, let t = 10 and solve for P(10).