Question

Population In Exercises \(51-54\) , the population (in millions) of a country in 2011 and the expected continuous annual rate of change \(k\) of the population are given. (Source: U.S. Census Bureau, International Data Base) Latvia \(\quad 2.2 \quad-0.006\)

Short answer

The population of Latvia after one year, with the given continuous rate of change, will be approximately 2.188 million.

Step-by-step solution

Understand the Variables

The given initial population \( P_0 \) is 2.2 million, the yearly rate of change \( k \) is -0.006 (indicating a decrease) and the target is to find the future population \( P \).

Apply the Continuous Growth Formula

To solve the problem, let's apply the continuous growth formula: \( P = P_0 e^{kt} \) . To find the population after one year, plug \( P_0 = 2.2, k = -0.006, t = 1 \) into the formula.

Solve for Population

Plug in the variables into the formula to compute the population after one year, which comes to \( P = 2.2 e^{(-0.006)(1)} \) . The computation gives the population to be approximately 2.188 million.

Key concepts

Continuous Growth Formula

In the world of exponential growth and decay, the continuous growth formula is a crucial tool. It helps us determine how a population or quantity changes over time given a certain percentage rate. The formula is expressed as:
\[ P = P_0 e^{kt} \]
where:

• \( P \): the future population or quantity,
• \( P_0 \): the initial population or quantity,
• \( k \): the continuous annual rate of change (negative for decay),
• \( t \): the time in years,
• \( e \): the base of the natural logarithm, approximately equal to 2.71828.

To solve a problem using this formula, identify your variables first (\( P_0 \), \( k \), \( t \)), and simply substitute them into the formula. This transforms the problem into an easy calculation. For example, with the given initial population of 2.2 million and a rate \( k \) of -0.006, you can predict future population sizes easily by substituting these values, as done in the original exercise.

Population Modeling

Population modeling is a method used to predict changes in population based on specific mathematical formulas. These models can help determine how a population grows or shrinks over time. In the case of Latvia's population, a continuous decay model is applied.
Such models often use exponential functions, such as the continuous growth formula. The natural growth or decline of a population is always influenced by certain parameters such as birth rates, death rates, and migration patterns. For Latvia, with a declining rate of -0.006, the population model reflects how the population decreases annually.
This type of modeling is crucial for planning in sectors like urban development, resource management, and economic forecasting. Policymakers use these insights to make informed decisions and create strategies that align with future population changes. By applying the continuous growth formula, we achieve a closer understanding of how current rates of change impact future populations.

Rate of Change

The rate of change in a population context describes how quickly or slowly a population increases or decreases. In mathematical terms, it's often represented by \( k \) in the continuous growth formula. A positive \( k \) indicates growth, whereas a negative \( k \) signifies decay.
In our exercise, Latvia's rate of change is -0.006, indicating a slight decline in population over time. This light decline translates into a gradual decrease in the number of people, providing insight into demographic trends.
Understanding the rate of change is essential for various applications:

• Governments can align their policies in areas like infrastructure and healthcare.
• Economists might predict market demands with this data in mind.
• Environmental agencies observe how populations impact ecosystems over time.

When using this concept, always remember that the continuous rate offers a seamless, consistent picture of change, smoothing out the random fluctuations that can occur in yearly data.