Final answer:
Using the law of exponential growth, we can determine that it will take approximately 7 years for the population of wolves to grow to 1000.
Step-by-step explanation:
To determine how long it will take for the population of wolves to grow to 1000, we can use the information provided to set up an exponential growth equation. Let's say that after t years, the population of wolves is given by P(t). We know that originally there were 200 wolves, and after 3 years, the population had grown to 270 wolves.
Using these initial conditions, we can set up the equation:
P(t) = P(0) * e^(kt)
where P(0) represents the initial population (200), e is the base of the natural logarithm (approximately 2.71828), k is the growth rate constant, and t is the time in years.
Now, we can solve for the growth rate constant, k:
270 = 200 * e^(3k)
Dividing both sides of the equation by 200 gives us:
1.35 = e^(3k)
Taking the natural logarithm of both sides of the equation:
ln(1.35) = 3k
Dividing both sides of the equation by 3:
k = ln(1.35) / 3
Now that we have the value of k, we can use it to determine the time it will take for the population to reach 1000 wolves:
1000 = 200 * e^((ln(1.35) / 3) * t)
Dividing both sides of the equation by 200:
5 = e^((ln(1.35) / 3) * t)
Taking the natural logarithm of both sides of the equation:
ln(5) = (ln(1.35) / 3) * t
Dividing both sides of the equation by ln(1.35) / 3:
t = (ln(5) * 3) / ln(1.35) ≈ 7.03
Therefore, it will take approximately 7 years for the population of wolves to grow to 1000.