The population of rabbits on an island is growing exponentially. In the year 2001, the population of rabbits was 2400, and by 2004 the population had grown to 2900. Predict the population of rabbits in the year 2012, to the nearest whole number.
To model the population growth of rabbits on the island, we can use the exponential growth formula:
P(t) = P0 * e^(kt)
Where P(t) is the population at time t, P0 is the initial population, e is the base of the natural logarithm (approximately 2.71828), k is the growth rate, and t is the time in years.
In this case, we know that the initial population in 2001 was 2400 (P0), and in 2004 the population had grown to 2900. We can use this information to find the growth rate (k).
2900 = 2400 * e^(3k)
Simplifying:
e^(3k) = 2900/2400
Taking the natural logarithm of both sides:
3k = ln(2900/2400)
k = ln(2900/2400) / 3
Using a calculator:
k ≈ 0.05902
Now we can use the growth rate (k) to predict the population in 2012:
P(2012) = 2400 * e^(0.05902 * (2012-2001))
Calculating:
P(2012) = 2400 * e^(0.05902 * 11)
P(2012) ≈ 2400 * e^(0.64922)
P(2012) ≈ 2400 * 1.915848
P(2012) ≈ 4598.036
Therefore, the population of rabbits in the year 2012, to the nearest whole number, is predicted to be 4598.