Answer:
To predict the population of rabbits in the year 2015, 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 constant.
Given that the population in 2005 (t = 0) was 6900, we have:
P(0) = 6900.
We're also given that by 2012 (t = 7), the population had grown to 13500, so we have:
P(7) = 13500.
We can use these two data points to solve for the growth rate constant, k.
Substituting the values into the formula:
13500 = 6900 * e^(k * 7).
Dividing both sides by 6900:
e^(k * 7) = 13500 / 6900.
Taking the natural logarithm of both sides:
k * 7 = ln(13500 / 6900).
Dividing both sides by 7:
k = ln(13500 / 6900) / 7.
Now that we have the value of k, we can predict the population in 2015 (t = 10) using the formula:
P(10) = P0 * e^(k * 10).
Substituting the values:
P(10) = 6900 * e^((ln(13500 / 6900) / 7) * 10).
Calculating this expression, we find:
P(10) ≈ 15711.
Therefore, the population of rabbits in the year 2015 is predicted to be approximately 15711 to the nearest whole number.
Hope that helps!