Problem 9 - 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.) Which forest's population is growing at a faster rate?

Step-by-Step Solution

Answer

Forest B's population is growing at a faster rate.

Step 1: Identify the Growth Rate

For both functions representing the forest populations, identify the base of the exponential function which represents the growth rate. In the function \(A(t) = 115(1.025)^t\), the growth rate is \(1.025\). For the function \(B(t) = 82(1.029)^t\), the growth rate is \(1.029\).

Step 2: Compare the Growth Rates

Compare the growth rates identified in Step 1. The base \(1.025\) from the function \(A(t)\) is smaller than \(1.029\) from the function \(B(t)\). Therefore, \(B(t)\) has a higher growth rate per year.

Step 3: Conclusion

Since the base of the exponential function for \(B(t)\) is larger than that for \(A(t)\), the population in forest B is growing at a faster rate than the population in forest A.

Key Concepts

Population Growth

Population growth refers to the way the number of individuals in a population increases over time. In the context of forests, it's about how the number of trees increases every year.
This happens due to several factors, including seed germination, favorable environmental conditions, and the absence of extreme conditions like wildfires.

• Natural conditions lead to fluctuations in tree populations.
• Stable and ideal conditions mean more predictable growth rates.
• Population growth models help predict future populations for resource management.

In our scenario, we analyze two neighboring forests, each having its tree population following its own growth pattern.
Each forest is represented by a unique exponential function, indicating differing growth patterns and external conditions.

Exponential Functions

Exponential functions are mathematical expressions used to model situations where a quantity grows or decreases at a consistent rate over time.
These functions have a constant base that dictates the rate at which growth or decay happens. They are represented in the form:
\[A(t) = a(b)^t\] where:

• \(a\) is the initial amount or population.
• \(b\) is the growth factor, more than 1 for growth, less than 1 for decay.
• \(t\) is time, usually in years.

In forest A and B:

• Function for forest A: \(A(t) = 115(1.025)^t\).
• Function for forest B: \(B(t) = 82(1.029)^t\).

The number inside the parentheses is crucial as it determines the growth rate.
A higher number means a faster growth rate, indicating a quicker increase in population.

Growth Rate Comparison

Comparing growth rates involves analyzing the bases of the exponential functions.
The base of each function directly reflects the growth rate:

• Forest A's growth rate: \(1.025\)
• Forest B's growth rate: \(1.029\)

When comparing these numbers, it's essential to note that even small differences can significantly affect population growth over time.
A base of \(1.029\) suggests that forest B's population increases each year by 2.9%, while forest A, at \(1.025\), grows by 2.5% annually.
Since \(1.029 > 1.025\), forest B grows at a faster rate.
Even a minor difference of 0.004 in these rates means forest B's population will exceed forest A's more quickly, showcasing how subtle variations in growth rates can lead to significant long-term effects.