Problem 10 - 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 had a greater number of trees initially? By how many?

Step-by-Step Solution

Answer

Forest A initially had 33 more trees than forest B.

Step 1: Identify Initial Populations

To determine which forest had a greater number of trees initially, we need to evaluate the functions \(A(t)\) and \(B(t)\) at \(t = 0\). When \(t = 0\), both functions represent their initial populations.

Step 2: Evaluate Initial Population for Forest A

Substitute \(t = 0\) into the function \(A(t) = 115(1.025)^t\):\[A(0) = 115(1.025)^0 = 115 \times 1 = 115\]The initial population of forest A is 115 trees.

Step 3: Evaluate Initial Population for Forest B

Substitute \(t = 0\) into the function \(B(t) = 82(1.029)^t\):\[B(0) = 82(1.029)^0 = 82 \times 1 = 82\]The initial population of forest B is 82 trees.

Step 4: Compare Initial Populations

Compare the results from A(0) and B(0): - Forest A has an initial population of 115 trees. - Forest B has an initial population of 82 trees. Forest A had a greater number of trees initially.

Step 5: Calculate the Difference in Initial Population

To find how many more trees forest A had than forest B initially, subtract the initial population of B from A:\[115 - 82 = 33\]Forest A initially had 33 more trees than forest B.

Key Concepts

Initial value

In the context of exponential growth, the initial value represents the starting quantity before any increase due to growth occurs. It acts as the anchor or the baseline for any growth equations you'll encounter. For population modeling, such as with the forests in this scenario, this starting point is crucial for understanding the initial conditions from which exponential changes unfold.

To determine the initial value for a population described by an exponential function, you simply evaluate the function at time zero, denoted by \(t = 0\). In both the forest scenarios provided, \(A(t)\) and \(B(t)\), substituting \(t = 0\) gives us:

• For \(A(t)\): \(A(0) = 115 \times (1.025)^0 = 115\)
• For \(B(t)\): \(B(0) = 82 \times (1.029)^0 = 82\)

By solving these, you can see that the initial population (or starting value) for Forest A is 115 trees, whereas for Forest B, it is 82 trees.

Population modeling

Population modeling is a powerful mathematical tool used to predict how populations change over time. An exponential function like those used here is particularly suited to modeling situations where the rate of growth is proportional to the present value, leading to the familiar curve that rises rapidly as time progresses.

In the functions provided:

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

The structure reflects a common form in exponential growth models: \(P(t) = P_0(1+r)^t\), where:

• \(P_0\) is the initial population value at \(t = 0\).
• \(r\) is the growth rate, here expressed as percentages (2.5% for Forest A, 2.9% for Forest B).
• \(t\) is the time, usually measured in years or other relevant increments.

This model highlights how even slight differences in growth rate can lead to significantly different population sizes over time, which is why accurate parameter estimation is essential.

Comparison of populations

Comparing populations in different growth scenarios allows us to explore various aspects of growth dynamics. In this exercise, the focus was on determining which forest started with a greater number of trees.

After calculating the initial values: 115 for Forest A and 82 for Forest B, it’s clear that Forest A had 33 more trees initially. However, besides initial numbers, you'll often be interested in growth rates as well. Even if one starts smaller, a higher growth rate can result in that population surpassing the other eventually.

For illustrative purposes:

• Forest A grows at 2.5% per year.
• Forest B grows at 2.9% per year.

This example reveals the nuance in "bigger isn't always better". While Forest A starts larger, Forest B's higher growth rate could eventually lead to surpassing Forest A, depending on the duration considered. Evaluating equations over time or at specific points helps make these comparisons concrete and informative.