Each year, a group of 150 Alaskan wolves has an average birth rate of 15% and an average death rate of 37% per year. Which function could be used to predict when there will be 25 wolves in this group? 150(1.15-0.37)^(t)=25, 150((1.15)^(t)-(0.63)^(t))=25, 150(1.15-0.63)^(t)=25, 150((1.15)^(t)-(0.37)^(t))=25
The Correct Answer and Explanation is:
Correct Answer:
150(1.15 – 0.37)ᵗ = 25
Explanation:
To determine the number of wolves in the group over time, we need to construct an exponential decay function that models the population change each year based on birth and death rates.
Step 1: Understand the Given Data
- Initial wolf population: 150
- Annual birth rate: 15% → This means the population increases by 15% each year
- Annual death rate: 37% → This means the population decreases by 37% each year
Since both processes (birth and death) are occurring simultaneously within the same year, we must combine them to determine the net population change per year.
Step 2: Calculate the Net Growth Rate
Net growth rate = Birth rate – Death rate
Net growth rate = 15% – 37% = –22%
This means the population is decreasing by 22% each year. In decimal form, the growth factor is:
1 – 0.22 = 0.78
So, each year, the population is multiplied by 0.78.
Step 3: Create the Exponential Model
An exponential decay function takes the form:
P(t) = P₀ × (r)ᵗ
Where:
- P(t) = population after t years
- P₀ = initial population (150)
- r = decay factor (0.78)
- t = number of years
Thus, the function becomes:
P(t) = 150 × (0.78)ᵗ
Now, we’re asked to find the equation that models when the population will decrease to 25 wolves, so we set:
150 × (0.78)ᵗ = 25
But note that 0.78 = 1.15 – 0.37, so this can also be written as:
150 × (1.15 – 0.37)ᵗ = 25
This exactly matches the first option:
👉 150(1.15 – 0.37)ᵗ = 25
Why the Other Options Are Incorrect:
- 150((1.15)ᵗ – (0.63)ᵗ) = 25 and others try to subtract exponential terms, which does not reflect a net rate of change and is mathematically incorrect for modeling combined birth-death effects.
- Only combining the birth and death rates before exponentiation gives a valid net growth model.
Final Answer:
✅ 150(1.15 – 0.37)ᵗ = 25