Problem 11 - College Algebra

Question

For the following exercises, consider this scenario: For each year \(t,\) the population of a forest of trees is represented by the function \(A(t)=115(1.025)^{t} .\) In a neighboring forest, the population of the same type of tree is represented by the function \(B(t)=82(1.029)^{t} .\) (Round answers to the nearest whole number.) Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after 20 years? By how many?

Step-by-Step Solution

Answer

Forest A will have 34 more trees than Forest B after 20 years.

Step 1: Define the Population Functions

We have two population functions: - For the first forest: \(A(t) = 115(1.025)^t\)- For the second forest: \(B(t) = 82(1.029)^t\)These functions represent the population of trees in each forest for year \(t\).

Step 2: Calculate Population of Forest A after 20 Years

Substitute \(t = 20\) into the function for forest A:\[ A(20) = 115(1.025)^{20} \]Calculate \(A(20)\) to find the population of forest A after 20 years:

Step 3: Evaluate A(20)

Calculate \((1.025)^{20}\):\(A(20) = 115 imes (1.025)^{20} \approx 188\)Thus, the population of forest A after 20 years is approximately 188 trees.

Step 4: Calculate Population of Forest B after 20 Years

Substitute \(t = 20\) into the function for forest B:\[ B(20) = 82(1.029)^{20} \]Calculate \(B(20)\) to find the population of forest B after 20 years:

Step 5: Evaluate B(20)

Calculate \((1.029)^{20}\):\(B(20) = 82 imes (1.029)^{20} \approx 154\)Thus, the population of forest B after 20 years is approximately 154 trees.

Step 6: Compare Populations

Now compare the populations: - Population of Forest A after 20 years: 188 trees - Population of Forest B after 20 years: 154 trees Forest A has more trees than Forest B.

Step 7: Calculate Difference

Subtract the population of Forest B from Forest A to find the difference: 188 (Forest A) - 154 (Forest B) = 34 trees. So, Forest A has 34 more trees than Forest B after 20 years.

Key Concepts

Population Modeling

Population modeling is an essential tool used to understand how populations of organisms, such as trees in a forest, grow over time. This type of modeling can involve simple or complex mathematical functions to depict real-world scenarios.
In our exercise, each forest's population grows exponentially, represented by specific functions with their distinct parameters. For Forest A, the function is given by \(A(t) = 115(1.025)^t\). This means that the initial population is 115 trees, and the population grows by a factor of 1.025 each year.

• Initial population: The constant before the exponential factor.
• Growth rate: The base of the exponent (1.025), indicating a 2.5% annual increase.

Similarly, for Forest B, the function \(B(t) = 82(1.029)^t\) indicates a starting population of 82 trees with a 2.9% annual increase. Through these models, we can simulate population changes over a designated period and make future predictions by evaluating these functions at specific time points.

Algebraic Functions

Algebraic functions are powerful tools in mathematics that help us express relationships between quantities. These functions can easily demonstrate the growth of a population when exponential growth is involved.
To understand how these functions operate within the context of population modeling, consider that both functions, \(A(t)\) and \(B(t)\), as exponential functions. They are characterized by a constant raised to the power affected by the variable time, \(t\).
This variable \(t\) indicates the passage of time in years in our problem. Exponential functions like these grow much faster than linear functions, as seen in their rapid increase over time. In our exercise, substituting a particular time value, such as \(t = 20\), into the function directly calculates tree populations after that number of years:

• For Forest A: \(A(20) = 115(1.025)^{20}\)
• For Forest B: \(B(20) = 82(1.029)^{20}\)

With algebraic manipulation and evaluation, we calculate an approximate number of trees for each forest after two decades, showing the power of algebraic expressions in modeling real-world scenarios.

Comparison of Growth Rates

Comparing growth rates is a crucial aspect when analyzing population models where exponential growth is involved. Exponential growth means the population increases by a constant rate relative to its current size, which significantly impacts long-term populations.
In the provided scenario, we compare two forests:

• Forest A with an annual growth rate of 2.5%
• Forest B with an annual growth rate of 2.9%

Even though Forest B has a higher percentage growth rate, Forest A started with a larger initial population. With exponential functions, the initial population and growth rate jointly define the population's future size. So while Forest B grows slightly faster in terms of percentage, its smaller starting size keeps its population lower than that of Forest A over 20 years.
After performing the calculations, we found that Forest A has 188 trees, and Forest B has 154 trees after 20 years. By subtracting these values, the difference in populations is 34 trees, indicating that despite faster growth, a higher initial population can lead to a larger final population size in the long term.