Problem 17 - Algebra 1: Concepts and Skills

Question

Write an exponential model to represent the situation. Tell what each variable represents. A population of \(100,000\) decreases by \(2 \%\) each year.

Step-by-Step Solution

Answer

The exponential decay model for this problem is \(P(t) = 100,000 * e^{-0.02t}\), where \(P(t)\) represents the population at year \(t\). The model assumes that the population decreases by \(2 \% \) each year from the initial population of \(100,000\).

Step 1: Understand the Problem

The given problem states that a population of \(100,000\) decreases by \(2 \%\) each year. The task is to write a mathematical model to represent this situation.

Step 2: Understand the Model

An exponential decay model is suitable for this kind of problem, since it involves a percentage decrease in population each year. The general form of an exponential decay model is \(P(t) = P_0 * e^{-rt}\), where:\n\n- \(P(t)\) is the population at time \(t\),\n- \(P_0\) is the initial population,\n- \(r\) is the decay rate, and\n- \(t\) is time.

Step 3: Define Variables

Express the given data in the problem in terms of the variables in the model.\n\n- \(P_0 = 100,000\), the initial population,\n- \(r = 2 \% = 0.02\), the annual decay rate as a decimal.

Step 4: Write the Model

Substitute the defined variables into the model to get the exponential decay model for this situation. So the model becomes \(P(t) = 100,000 * e^{-0.02t}\).

Step 5: Interpret the Variables

This model computes the population \( P(t) \) at time year \( t \), given an initial population of \( 100,000 \) and annual decay rate of \( 2 \% \).

Key Concepts

Exponential Model

An exponential model is a mathematical representation frequently used to describe situations where a quantity changes at a constant percentage rate per unit of time. In this case, it becomes a convenient tool to analyze how a population decreases yearly. The general equation for exponential decay is given by:

• \( P(t) = P_0 \cdot e^{-rt} \)

This formula helps anticipate future amounts by considering the initial amount, the decay rate, and time.
In our example, the population starts at \( 100,000 \), reducing annually by \( 2 \% \). The exponential function includes the decay rate normally expressed in its decimal form, here \( 0.02 \). By applying these values, we generate a model that accurately depicts the population dynamics over time, providing predictions without manually calculating each year's decrement.

Population Decay

Population decay describes a reduction in the number of individuals within a group over time, often due to death, migration, or other factors. It's an essential concept in ecology and demography, used to plan resources and understand environmental impacts.
In the context of this model, population decay is expressed through the decay rate \( r \). It signifies the percentage by which the population decreases each year. A \( 2 \% \) decay rate means the population loses \( 2 \% \) of its members annually. The exponential model calculates the population for any year \( t \) effectively capturing this continuous decrease.
Understanding how quickly a population depletes is key in planning and intervention strategies. Exponential models provide insights into how soon a population might reach critical levels.

Mathematical Modeling

Mathematical modeling is the use of mathematical expressions to depict real-world situations, allowing predictions and deeper insights. Models simplify complex systems, making them easier to study and understand.
In this example of exponential decay, mathematical modeling translates a dynamic population situation into a clear equation. Each variable represents a real-world factor:

• Initial population \( P_0 \)
• Decay rate \( r \)
• Time \( t \)

By substituting real values, we gain the ability to project changes over time. Models not only predict future scenarios but also guide policy and decision-making. They are invaluable in fields such as ecology, economics, and engineering.