Problem 11 - 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 growth of the forest will have a greater number of trees after 20 years? By how many?

Step-by-Step Solution

Answer

Forest A has 53 more trees than Forest B after 20 years.

Step 1: Evaluate the population of Forest A after 20 years

To find the population of forest A after 20 years, use the formula \(A(t) = 115(1.025)^t\) with \(t = 20\). Calculate \(A(20) = 115(1.025)^{20}\). Use a calculator to find the result for \(1.025^{20}\), then multiply by 115: \(A(20) \approx 115 \times 1.6436 \approx 189\) (rounded to the nearest whole number).

Step 2: Evaluate the population of Forest B after 20 years

Use the formula \(B(t) = 82(1.029)^t\) to find the population of forest B after 20 years by setting \(t = 20\). Calculate \(B(20) = 82(1.029)^{20}\). Use a calculator to find \(1.029^{20}\), then multiply by 82: \(B(20) \approx 82 \times 1.6598 \approx 136\) (rounded to the nearest whole number).

Step 3: Compare the populations

Compare the results from Step 1 and Step 2 to determine which forest has more trees. Forest A has \(189\) trees and Forest B has \(136\) trees after 20 years.

Step 4: Calculate the difference in population between the two forests

Subtract the population of Forest B from that of Forest A to find out how many more trees Forest A has: \(189 - 136 = 53\). Forest A has 53 more trees than Forest B after 20 years.

Key Concepts

Population Modeling

Population modeling is an essential concept in understanding any kind of growth, whether it be biological, ecological, or even economic. In this exercise, we observe exponential functions that model tree populations in two different forests. Exponential functions are used because they effectively represent natural growth processes. The formula used here is of the form \(P(t) = P_0(1 + r)^t\), where:

• \(P(t)\) is the population at time \(t\).
• \(P_0\) is the initial population.
• \(r\) is the growth rate.
• \(t\) is the time in years.

Here, Forest A starts with 115 trees and grows at a rate of 2.5% yearly, while Forest B begins with 82 trees and grows at a slightly higher rate of 2.9% annually. By plugging in the value of \(t = 20\) years into each respective formula, we can predict how many trees each forest will have in 20 years. Understanding these basic components helps us model the population accurately and foresee possible future scenarios.

Comparison of Growth Rates

Growth rates can significantly affect the future size of populations as shown in this exercise. Even a slight difference in growth rate can lead to a notable difference over time. Forest A has a growth rate of 2.5% while Forest B enjoys a slightly faster rate of 2.9%. Despite this, due to the larger initial number of trees, Forest A surpasses Forest B in tree population after 20 years. To better compare, we analyzed:

• Forest A after 20 years: \(A(20) = 115(1.025)^{20} \approx 189\)
• Forest B after 20 years: \(B(20) = 82(1.029)^{20} \approx 136\)

Comparing these results shows Forest A has 53 more trees than Forest B at the end of 20 years, demonstrating how initial values combined with growth rates influence eventual outcomes.

Forest Population Dynamics

Forest population dynamics are complex and influenced by various factors, with growth rates and initial populations being primary factors as demonstrated in this exercise. Dynamic systems like forests can be modeled to some extent using mathematical equations, providing insights into the long-term sustainability and trends of populations. Here are some factors that can affect the population dynamics of forests:

• Growth rate changes due to environmental factors or genetic variations.
• Initial population size and healthy reproduction rates.
• Human activities such as logging and reforestation.
• Natural events like wildfires or disease outbreaks.

In our example, while the growth models predict different outcomes based on initial data, real-world situations would require further considerations including these dynamic factors.