Let's analyze the problem step-by-step to determine which function could be used to predict when there will be 25 wolves in this group and solve for the time [tex]\( t \)[/tex].
### Step 1: Understanding the Population Dynamics
1. Initial Population: The initial population of the Alaskan wolves is 150.
2. Birth Rate: The average birth rate is [tex]\(15\%\)[/tex] per year.
3. Death Rate: The average death rate is [tex]\(37\%\)[/tex] per year.
### Step 2: Determine the Net Growth Rate
To find the net growth rate, we subtract the death rate from the birth rate:
[tex]\[
\text{Net Growth Rate} = 15\% - 37\% = -22\%
\][/tex]
Since the net growth rate is [tex]\(-22\%\)[/tex], the population decreases by [tex]\(22\%\)[/tex] each year.
### Step 3: Populating the Exponential Decay Model
The appropriate model for this population decrease can be represented as:
[tex]\[
P(t) = P_0 \times (1 + r)^t
\][/tex]
where:
- [tex]\(P(t)\)[/tex] is the population at time [tex]\(t\)[/tex].
- [tex]\(P_0\)[/tex] is the initial population (150 wolves).
- [tex]\(r\)[/tex] is the net growth rate [tex]\((-22\%\)[/tex] or [tex]\(-0.22\)[/tex]).
- [tex]\(t\)[/tex] is the time in years.
Since [tex]\( r = -0.22 \)[/tex], the base of the exponential function becomes [tex]\(1 - 0.22 = 0.78\)[/tex].
### Step 4: Setting Up the Equation
We need to find out when the population will be 25 wolves. Thus, we set up the equation:
[tex]\[
150 \times (0.78)^t = 25
\][/tex]
### Step 5: Solving the Equation
To solve for [tex]\(t\)[/tex]:
1. First, isolate [tex]\( (0.78)^t \)[/tex]:
[tex]\[
(0.78)^t = \frac{25}{150}
\][/tex]
[tex]\[
(0.78)^t = \frac{1}{6}
\][/tex]
2. Now take the natural logarithm of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[
\ln((0.78)^t) = \ln\left(\frac{1}{6}\right)
\][/tex]
Using the property of logarithms [tex]\( \ln(a^b) = b \ln(a) \)[/tex]:
[tex]\[
t \ln(0.78) = \ln\left(\frac{1}{6}\right)
\][/tex]
3. Finally, solve for [tex]\( t \)[/tex]:
[tex]\[
t = \frac{\ln\left(\frac{1}{6}\right)}{\ln(0.78)}
\][/tex]
### Step 6: Calculation (Numerically Solving)
Solving this numerically (we already have the result given in the problem):
[tex]\[
t \approx 7.211
\][/tex]
### Conclusion
The function we used to predict the population and the time it takes for the population to decline to 25 wolves is:
[tex]\[
150 \times (0.78)^t = 25
\][/tex]
And solving this equation gives us [tex]\( t \approx 7.211 \)[/tex] years. Thus, it will take approximately 7.211 years for the group of 150 Alaskan wolves to decrease to 25 wolves with the given birth and death rates.
