Problem 12 - Precalculus

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 forest will have a greater number of trees after 100 years? By how many?

Step-by-Step Solution

Answer

The first forest will have 64 more trees after 100 years.

Step 1: Understand the Problem

We need to determine which forest will have more trees after 100 years using the population growth models. Given functions are: \( A(t) = 115(1.025)^t \) for the first forest and \( B(t) = 82(1.029)^t \) for the second forest.

Step 2: Substitute t = 100 into A(t)

We calculate the population of trees in the first forest after 100 years using the formula for \( A(t) \):\[ \begin{align*} A(100) &= 115 \times (1.025)^{100} \end{align*} \]Substitute 100 into the exponent and calculate.

Step 3: Calculate A(100)

Using a calculator for precise computation, find \( (1.025)^{100} \), and multiply by 115:\[ \begin{align*} A(100) &\approx 115 \times 10.873 \approx 1249.395 \ \end{align*} \]Round this to the nearest whole number. Thus, \( A(100) \approx 1249 \).

Step 4: Substitute t = 100 into B(t)

Now calculate the population of trees in the second forest after 100 years using the formula for \( B(t) \):\[ \begin{align*} B(100) &= 82 \times (1.029)^{100} \end{align*} \]

Step 5: Calculate B(100)

Using a calculator, find \( (1.029)^{100} \) and multiply by 82:\[ \begin{align*} B(100) &\approx 82 \times 14.446 \approx 1184.572 \end{align*} \]Round to the nearest whole number. Hence, \( B(100) \approx 1185 \).

Step 6: Compare the Populations

The population of the first forest after 100 years is approximated to 1249, and the second forest to 1185. Therefore, the first forest has more trees.

Step 7: Calculate the Difference

Find the difference in tree populations between the first and second forest after 100 years:\[ 1249 - 1185 = 64 \]The first forest has 64 more trees.

Key Concepts

Population Growth Models

Population growth models are mathematical expressions used to predict population changes over time. These models are crucial in understanding how populations of living organisms, such as trees in a forest, will evolve. There are two primary types of growth models often referred to in ecology:

• Exponential Growth Models
• Logistic Growth Models

The exponential growth model is characterized by steady and continuous growth. It assumes that resources are abundant, allowing the population to grow without limits. This model is represented by functions like:\[ P(t) = P_0 imes (1 + r)^t \]where \( P(t) \) is the population at time \( t \), \( P_0 \) is the initial population, and \( r \) is the growth rate.
Each forest in the problem statement uses an exponential growth model. The formula descriptions, \( A(t) = 115(1.025)^t \) and \( B(t) = 82(1.029)^t \), show differences in initial populations and growth rates. Understanding how these values influence growth helps predict future population sizes.

Forest Population Comparison

To compare forest populations using mathematical models, it's vital to analyze and compare the parameters of each model. The parameters include the initial population size and the growth rate. Let's use our current example to understand this comparison:

• Forest A: \( A(t) = 115(1.025)^t \)
• Forest B: \( B(t) = 82(1.029)^t \)

The initial population for Forest A is 115, and for Forest B, it's 82. Despite starting with fewer trees, Forest B has a slightly higher growth rate of 2.9% per year compared to Forest A's 2.5%.
At the end of 100 years, we calculated that Forest A had approximately 1249 trees, while Forest B had around 1185 trees. Conclusively, despite its slower growth rate and larger initial population, Forest A outnumbers Forest B in the long-term growth comparison. This reveals that the compound effect over time at even slightly different growth rates can greatly influence total population.

Growth Rate Calculation

The growth rate is a crucial element in understanding population dynamics. It determines how quickly a population expands over a period. Here's how you can calculate and interpret growth rates in exponential growth models:
The growth rate in percentage form can often be found in a growth model represented as \((1 + r)\), where \( r \) corresponds to the decimal growth rate.To calculate the population after a specific period using a given growth rate:

1. Identify the initial quantity (e.g., the initial number of trees).
2. Use the given growth rate to project future populations using the formula: \[ P(t) = P_0 imes (1 + r)^t \]

In the given scenarios:

• Forest A grows by 2.5% annually, represented as \( 1.025 \) in the model.
• Forest B grows by 2.9% annually, denoted by \( 1.029 \).

Through calculations, these small differences in growth rates lead to significant differences in population numbers over 100 years. Understanding and applying this concept is essential for predicting long-term outcomes based on growth trends.