The population of rabbits on an island is growing exponentially. In the year 1995, the population of rabbits was 8600, and by 1997 the population had grown to 10900. Predict the population of rabbits in the year 2007, to the nearest whole number.
To predict the population of rabbits in the year 2007, we need to use the exponential growth formula:
P(t) = P0 * (1 + r)^t
Where:
- P(t) is the predicted population at time t
- P0 is the initial population
- r is the growth rate
- t is the time period
Given that the population in 1995 (P0) is 8600 and the population in 1997 is 10900, we can calculate the growth rate (r) as follows:
10900 = 8600 * (1 + r)^2
Divide both sides of the equation by 8600:
1.2674 = (1 + r)^2
The value of (1 + r) is approximately equal to the square root of 1.2674, which is approximately 1.1266 (to four decimal places). Subtracting 1 from this value gives us the growth rate, r:
r = 1.1266 - 1 = 0.1266
Now we need to calculate the population in 2007 (t = 2007 - 1995 = 12):
P(12) = 8600 * (1 + 0.1266)^12
Using a calculator or spreadsheet, we can determine that (1 + 0.1266)^12 is approximately equal to 2.1024 (to four decimal places):
P(12) = 8600 * 2.1024 = 18090.0
Rounding to the nearest whole number gives us a predicted population of 18090 rabbits in the year 2007.