Answer:
a) Let P(t) be the population of the colony at time t in years. Since the population grows exponentially, we can model the population with the formula P(t) = P0e^kt, where P0 is the initial population (when t=0), and k is a constant. We know that the colony begins with 18 rabbits, so P0 = 18. We also know that five years later, the population is 300, so P(5) = 300. Plugging in these values, we get:
300 = 18e^(5k)
Dividing both sides by 18 and taking the natural logarithm of both sides, we get:
ln(300/18) = 5k
Simplifying, we get:
k = ln(300/18) / 5
So the formula for the population of the colony is:
P(t) = 18e^(t*ln(300/18)/5)
b) To estimate how long it takes for the population to reach 2000 rabbits, we need to solve for t in the equation P(t) = 2000. Plugging in the formula from part (a), we get:
2000 = 18e^(t*ln(300/18)/5)
Dividing both sides by 18 and taking the natural logarithm of both sides, we get:
ln(2000/18) = t*ln(300/18)/5
Simplifying, we get:
t = 5*ln(2000/18) / ln(300/18)
Using a calculator, we get:
t ≈ 23.7 years
So it will take about 23.7 years for the population to reach 2000 rabbits.
Explanation: