Problem 70 - Precalculus : Building Concepts and Connections

Question

Fill in the table according to the given rule and find an expression for the function represented by the rule. A population of cockroaches starts out at 100 and doubles every month. $$\begin{aligned} &\begin{array}{cc} \text { Month } & \text { Population } \end{array}\\\ &\begin{array}{l} 0 \\ 1 \\ 2 \\ 3 \\ 4 \end{array} \end{aligned}$$

Step-by-Step Solution

Answer

The completed table is {(0,100),(1,200),(2,400),(3,800),(4,1600)}. The function that describes this is \( P = 100 \times 2^{m}\)

Step 1: Fill the Table

The initial cockroach population of 100 doubles each month. In the first month, there will be \(100\times2=200\), in the second month \(200\times 2 = 400\), in the third month, \(400 \times 2 = 800\), and finally in the fourth month \(800 \times 2 = 1600\).

Step 2: Identify the Function

This process of doubling the population each month is an example of an exponential function. Here the base is 2 (as the population doubles), and the exponent corresponds to the month.

Step 3: Formulate the Function

The function that describes this situation is \( P = 100 \times 2^{m}\) where m is the month number and P is the cockroach population.

Key Concepts

Population Growth

Population growth is the increase in the number of individuals in a population over time. In our example, a population of cockroaches starts at 100 and sees growth by doubling each month. This showcases natural, biological growth which can often be rapid in species like insects. Population growth can be influenced by various factors including birth rates and environmental conditions which determine the capacity to support new life. It's important to consider how growth rates can determine the eventual size of a population and how it impacts resources. Little changes in growth rates can lead to significant population increases or decreases over time.

Exponential Growth

Exponential growth is a pattern of data that shows greater increases over time, creating a J-shaped curve when graphed. It can be identified when the increase rate of a population becomes more rapid in proportion to the growing total number. For cockroaches, starting with an initial amount of 100, the growth doubles each month (100, 200, 400, 800, and 1600).

Key characteristics of exponential growth include:

• Constant doubling time: The time it takes for the population to double remains constant.
• Rapid increase: Growth accelerates swiftly as the population base expands.
• Doubling base: In our example, each successive month sees a doubling of the existing population.

Over time, this type of growth can lead to extraordinary numbers if unchecked, as shown with our continuously doubling cockroach population.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent real-world scenarios, such as population growth in this problem. In the exercise, the population can be represented with an exponential function, which aptly models the doubling process.

The function used is: \( P = 100 \times 2^{m} \), where:

• \( P \) represents the population at a given month.
• \( m \) is the month number.
• The base 2 is chosen since the population doubles.

Mathematical modeling in this context provides a clear and precise way to predict population sizes in future months, making it a powerful tool in both biological and environmental studies. By understanding these functions, scientists can forecast outcomes and make informed decisions related to ecosystems and resource management.