9. Wolves were removed from Yellowstone National Park in 1926. After the wolves were gone, scientists discovered that the entire park was much healthier when there were wolves in the park. In 1995, scientists reintroduced the 8 wolves into Yellowstone National Park. The wolf population in Yellowstone National Park grows exponentially at a rate of 35% per year. The scientists can't let the wolf population be larger than 200. What year will there be more than 200 wolves in the park?
9. Wolves were removed from Yellowstone National Park in 1926. After the wolves were gone, scientists discovered that the entire park was much healthier when there were wolves in the park. In 1995, scientists reintroduced the 8 wolves into Yellowstone National Park. The wolf population in Yellowstone National Park grows exponentially at a rate of 35% per year. The scientists can't let the wolf population be larger than 200. What year will there be more than 200 wolves in the park?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Yellowstone Wolf Reintroduction: A Timeline and Population Growth Analysis**
In 1926, wolves were removed from Yellowstone National Park. Following their removal, scientists observed that the park's overall health declined, highlighting the importance of wolves in the ecosystem. Subsequently, in 1995, scientists reintroduced 8 wolves into Yellowstone National Park.
Since their reintroduction, the wolf population in Yellowstone has been growing exponentially at a rate of 35% per year. Given this growth rate, scientists aim to manage the population to ensure it does not exceed 200 wolves.
### Problem Statement:
**Question**: Based on the exponential growth rate of 35% per year, in what year will the number of wolves in the park exceed 200?
### Solution:
1. **Initial Population (1995)**: 8 wolves
2. **Growth Rate**: 35% per year
We use the formula for exponential growth:
\[ P(t) = P_0 (1 + r)^t \]
Where:
- \( P(t) \) is the wolf population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the growth rate,
- \( t \) is the time in years since the initial population.
Here, \( P_0 = 8 \) wolves, \( r = 0.35 \), and we need to find \( t \) when \( P(t) > 200 \).
\[ 200 = 8 (1.35)^t \]
\[ \frac{200}{8} = (1.35)^t \]
\[ 25 = (1.35)^t \]
Taking the natural logarithm of both sides:
\[ \ln(25) = t \ln(1.35) \]
\[ t = \frac{\ln(25)}{\ln(1.35)} \]
Calculating the values:
\[ t = \frac{3.2189}{0.3001} \approx 10.73 \]
Therefore, it will take approximately 11 years from 1995 for the wolf population to exceed 200.
**Conclusion**: More than 200 wolves are expected to inhabit Yellowstone National Park by the year 2006. This timeline emphasizes the significance of managing the wolf population to balance the ecosystem's health while avoiding overpopulation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5b1aebc2-0755-46cf-ac8d-ba71bc9db631%2F788c1129-0d59-4c37-abd0-148d29efe901%2Fvb32u82_processed.png&w=3840&q=75)
Transcribed Image Text:**Yellowstone Wolf Reintroduction: A Timeline and Population Growth Analysis**
In 1926, wolves were removed from Yellowstone National Park. Following their removal, scientists observed that the park's overall health declined, highlighting the importance of wolves in the ecosystem. Subsequently, in 1995, scientists reintroduced 8 wolves into Yellowstone National Park.
Since their reintroduction, the wolf population in Yellowstone has been growing exponentially at a rate of 35% per year. Given this growth rate, scientists aim to manage the population to ensure it does not exceed 200 wolves.
### Problem Statement:
**Question**: Based on the exponential growth rate of 35% per year, in what year will the number of wolves in the park exceed 200?
### Solution:
1. **Initial Population (1995)**: 8 wolves
2. **Growth Rate**: 35% per year
We use the formula for exponential growth:
\[ P(t) = P_0 (1 + r)^t \]
Where:
- \( P(t) \) is the wolf population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the growth rate,
- \( t \) is the time in years since the initial population.
Here, \( P_0 = 8 \) wolves, \( r = 0.35 \), and we need to find \( t \) when \( P(t) > 200 \).
\[ 200 = 8 (1.35)^t \]
\[ \frac{200}{8} = (1.35)^t \]
\[ 25 = (1.35)^t \]
Taking the natural logarithm of both sides:
\[ \ln(25) = t \ln(1.35) \]
\[ t = \frac{\ln(25)}{\ln(1.35)} \]
Calculating the values:
\[ t = \frac{3.2189}{0.3001} \approx 10.73 \]
Therefore, it will take approximately 11 years from 1995 for the wolf population to exceed 200.
**Conclusion**: More than 200 wolves are expected to inhabit Yellowstone National Park by the year 2006. This timeline emphasizes the significance of managing the wolf population to balance the ecosystem's health while avoiding overpopulation.
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