The population of rabbits on an island is growing exponentially. In the year 1992, the population of rabbits was 8300, and by 2000 the population had grown to 21500. Predict the population of rabbits in the year 2009, to the nearest whole number.
To solve this problem, we can use the formula for exponential growth:
P(t) = P₀ * e^(kt)
Where:
P(t) = Population at time t
P₀ = Initial population
e = Euler's number (approximately 2.71828)
k = Growth rate constant
t = Time (in this case, measured in years)
We can find the growth rate constant (k) by using the information given for the years 1992 and 2000. Let's call 1992 "t₀" and 2000 "t₁":
P(t₁) = P₀ * e^(k*t₁)
21500 = 8300 * e^(k*8)
Dividing both sides of the equation by 8300, we get:
2.59036145 = e^(k*8)
Taking the natural logarithm of both sides of the equation, we have:
ln(2.59036145) = k*8
Solving for k:
k = ln(2.59036145) / 8
Using a calculator, we find that k is approximately 0.080513.
Now that we have the value of k, we can determine the population in 2009 (t = 17). Plugging the values into the exponential growth formula:
P(t) = P₀ * e^(kt)
P(17) = 8300 * e^(0.080513*17)
P(17) ≈ 8300 * e^1.370221
Using a calculator to evaluate the expression, we find that P(17) is approximately 57,913.
Therefore, the predicted population of rabbits in the year 2009 is 57,913.