A native wolf species has been reintroduced into a national forest. Originally 200
wolves were transplanted. After 3 years, the population had grown to 270 wolves. If
the population grows exponentially,
a. Write a recursive formula for the number of wolves
b. Write an explicit formula for the number of wolves
c. If this trend continues, how many wolves will there be in 10 years?
a. To write a recursive formula for the number of wolves, we need to identify the growth rate. Let's assume that the wolf population grows by a rate of 20% annually.
At the starting population (n = 0), there were 200 wolves.
In the subsequent year (n = 1), the population grew by 20%, giving us 200 + 20% of 200 = 240 wolves.
In the next year (n = 2), the population grew by 20% again, giving us 240 + 20% of 240 = 288 wolves.
Generalizing this pattern, we can write the recursive formula:
P(n) = P(n-1) + r * P(n-1)
Where P(n) is the population at year n, P(n-1) is the population at year n-1, and r is the growth rate.
b. An explicit formula can be derived from the recursive formula. For an exponential growth scenario, the explicit formula looks like this:
P(n) = P(0) * (1 + r)^n
Where P(0) is the initial population, r is the growth rate, and n is the number of years.
In this case, the explicit formula is:
P(n) = 200 * (1 + 0.2)^n
c. To find the number of wolves in 10 years using the explicit formula, we substitute n = 10:
P(10) = 200 * (1 + 0.2)^10
P(10) ≈ 200 * 1.485997
P(10) ≈ 297.1994
Therefore, if the trend continues, there will be approximately 297 wolves in 10 years.
To find the recursive formula for the number of wolves, we need to determine the growth rate. In this case, since the population is growing exponentially, we can assume a constant growth rate.
Let's assume the growth rate is represented by the letter "r." We know that after 3 years, the population has grown to 270 wolves from the initial 200 wolves. We can use this information to find the value of "r."
The recursive formula to represent the number of wolves is as follows:
N(n) = N(n-1) + r * N(n-1)
Where:
N(n) represents the number of wolves at year n,
N(n-1) represents the number of wolves at year n-1,
and r represents the growth rate.
In this case, we know that N(3) is 270, and N(3-1) is 200. Plugging in the values, we have:
270 = 200 + r * 200
To solve for r, we can rearrange the equation:
r = (270 - 200) / 200
r = 70 / 200
r = 0.35
So the recursive formula for the number of wolves is:
N(n) = N(n-1) + 0.35 * N(n-1)
Now let's move on to finding the explicit formula.
To find the explicit formula for the number of wolves, we need an initial condition. In this case, we know that after 3 years, the population is 270 wolves.
The explicit formula to represent the number of wolves is as follows:
N(n) = N(0) * (1 + r)^n
Where:
N(n) represents the number of wolves at year n,
N(0) represents the initial number of wolves,
r represents the growth rate,
and n represents the number of years.
Using the initial condition of N(3) = 270 and the growth rate of r = 0.35, we can plug in the values and solve for N(0):
270 = N(0) * (1 + 0.35)^3
Simplifying the equation:
270 = N(0) * 1.453125
To solve for N(0):
N(0) = 270 / 1.453125
N(0) ≈ 185.714
So the explicit formula for the number of wolves is:
N(n) ≈ 185.714 * (1 + 0.35)^n
Finally, to find the number of wolves in 10 years (N(10)), we can plug in the values into the explicit formula:
N(10) ≈ 185.714 * (1 + 0.35)^10
N(10) ≈ 185.714 * 2.208478751
N(10) ≈ 410.741
Therefore, if the trend continues, there will be approximately 410 wolves in the national forest after 10 years.