Problem 17 - Algebra A Combined Function

Question

Solve. Unless noted otherwise, round answers to the nearest whole. National Park Service personnel are trying to increase the size of the bison population of Theodore Roosevelt National Park. If 260 bison currently live in the park, and if the population's rate of growth is \(2.5 \%\) annually, find how many bison there should be in 10 years.

Step-by-Step Solution

Answer

There will be approximately 333 bison in 10 years.

Step 1: Understand the Problem Statement

The problem provides that 260 bison currently inhabit the park and their population grows at a rate of 2.5% per year. We are asked to find out how many bison will be there after 10 years.

Step 2: Identify the Formula

To solve this, we need the formula for exponential growth: \( P = P_0 (1 + r)^t \), where \( P \) is the future population, \( P_0 \) is the initial population, \( r \) is the growth rate (as a decimal), and \( t \) is the time in years.

Step 3: Substitute the Given Values

Using the formula, substitute: \( P_0 = 260 \), \( r = 0.025 \), and \( t = 10 \). So, the formula becomes: \( P = 260 (1 + 0.025)^{10} \).

Step 4: Calculate the Growth Factor

First, calculate the growth factor: \( 1 + 0.025 = 1.025 \).

Step 5: Raise the Growth Factor to the Power of the Time

Now, compute \( 1.025^{10} \). You can use a calculator to find that \( 1.025^{10} \approx 1.28008 \).

Step 6: Calculate the Future Population

Multiply the initial population by the computed growth factor: \( P = 260 \times 1.28008 \).

Step 7: Round the Result

Calculate the product: \( 260 \times 1.28008 \approx 332.82 \). Round this number to the nearest whole number to get 333.

Key Concepts

Bison Population

The bison population in natural reserves like national parks is crucial for maintaining ecological balance. It serves as a vital part of the ecosystem, influencing many plant and animal species.

• Each bison contributes to the biological diversity of the area, promoting native flora through natural grazing.
• Their presence is often utilized as an indicator of ecosystem health and biodiversity.
• For conservationists, maintaining a sustainable bison population is a primary goal, as it ensures the preservation of the natural environment.

To predict how a population will change over time, simple models using exponential growth help estimate future numbers.
Making these predictions involves understanding initial population counts and their expected rate of growth.

Theodore Roosevelt National Park

Theodore Roosevelt National Park, located in North Dakota, is one of the few natural habitats preserved for bison in the United States. It's named after the 26th U.S. President, who was known for his role in conservation.

• As a place rich in history and natural beauty, the park serves as a sanctuary for a range of wildlife, including the majestic bison.
• The park not only aims to protect these creatures but also offers educational programs to enhance public understanding.
• Maintaining bison populations here is part of broader efforts to restore these animals to their historic range.

The park employs various strategies, such as controlled growth and habitat management, to support the flourishing of the bison population.
Rangers and scientists regularly monitor these strategies to ensure the animals continue to thrive in a changing environment.

Growth Rate

A growth rate is a valuable metric in biology and ecology to predict changes in population over time. It is especially important in determining how species like the bison will fare in their habitats.

• The growth rate provides insight into how quickly the population can expand given the available resources and environmental conditions.
• In this scenario, the bison population is growing at a rate of 2.5% per year, which is a moderate rate suggesting healthy growth.
• Exponential growth models help visualize the impact of this steady percentage increase over a specific period.

By calculating the expected future population using the formula \( P = P_0 (1 + r)^t \), we predict how population trends develop under current conditions.
This helps in setting conservation targets and ensures that survival and quality of life for the species are prioritized.