Problem 81 - Intermediate Algebra

Question

Highs and Lows. Kuwait, located at the head of the Persian Gulf, has one of the greatest population growth rates in the world. Bulgaria, in southeastern Europe, has one of the smallest. Use an exponential growth/decay model to complete the table. $$ \begin{array}{|l|c|c|c|} \hline & & \text { Annual } & \text { Estimated } \\ \text { Country } & \begin{array}{c} \text { Population } \\ \text { 2010 } \end{array} & \begin{array}{c} \text { growth } \\ \text { rate } \end{array} & \begin{array}{c} \text { population } \\ \mathbf{2 0 2 5} \end{array} \\ \hline \text { Kuwait } & 2,789,132 & 3.501 \% & \\ \hline \text { Bulgaria } & 7,148,785 & -0.768 \% & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

Kuwait's estimated 2025 population is 4,708,854 and Bulgaria's is 6,336,505.

Step 1: Understand Exponential Growth/Decay Formula

To estimate future population using exponential growth or decay, we use the formula: \( P = P_0 (1 + r)^t \), where \( P \) is the future population, \( P_0 \) is the initial population size, \( r \) is the growth rate (as a decimal), and \( t \) is the time in years.

Step 2: Calculate Kuwait's Population in 2025

For Kuwait, we have \( P_0 = 2,789,132 \), \( r = 0.03501 \), and \( t = 15 \) years (2025 - 2010). Use the formula: \[ P = 2,789,132 \times (1 + 0.03501)^{15} \]. Calculate \( (1 + 0.03501)^{15} \), and then multiply by 2,789,132 to find the estimated population.

Step 3: Calculate Bulgaria's Population in 2025

For Bulgaria, we have \( P_0 = 7,148,785 \), \( r = -0.00768 \), and \( t = 15 \). Use the formula: \[ P = 7,148,785 \times (1 - 0.00768)^{15} \]. Calculate \( (1 - 0.00768)^{15} \) and then multiply by 7,148,785 to find the estimated population.

Step 4: Compute Kuwait's Future Population

After calculating, \( (1 + 0.03501)^{15} \approx 1.6881 \). Multiply this by Kuwait's initial population: \[ P = 2,789,132 \times 1.6881 \approx 4,708,854 \].

Step 5: Compute Bulgaria's Future Population

After calculating, \( (1 - 0.00768)^{15} \approx 0.8866 \). Multiply this by Bulgaria's initial population: \[ P = 7,148,785 \times 0.8866 \approx 6,336,505 \].

Key Concepts

Population Growth Rate

The population growth rate is essential in understanding how quickly a population increases or decreases over time. It is usually expressed as a percentage and can be positive for growth or negative for decline.
In our example, Kuwait has a population growth rate of 3.501%, indicating a rapid increase in its population. On the other hand, Bulgaria's population shows a decline with a growth rate of -0.768%.
To work with growth rates mathematically, it's important to convert them to decimals. For instance, Kuwait's rate becomes 0.03501 when expressed as a decimal.
For exponential models, this rate provides a clear, consistent basis for projecting future changes in population size over a specific period. Understanding the growth rate is crucial for predicting trends and making informed decisions regarding resources and planning.

Mathematical Modeling

Mathematical modeling with exponential growth or decay helps quantify and predict changes in population over time.
The formula used in such models is \( P = P_0 (1 + r)^t \), where

• \( P \) is the future population,
• \( P_0 \) is the current or initial population,
• \( r \) is the growth rate in decimal form,
• \( t \) represents the number of years into the future.

Exponential growth reflects a situation where the population grows by a consistent percentage over each time period. In contrast, exponential decay illustrates a steady decline.
This model is fundamental in various fields like ecology, demographics, and economics, offering a reliable way to understand complex systems and predict future scenarios.

Future Population Estimate

Estimating future populations involves applying the exponential growth or decay model to the specific context.
For Kuwait, with its high growth rate, estimating the population in 2025 involves projecting forward from a base year of 2010. The calculation, \[ P = 2,789,132 \times (1 + 0.03501)^{15} \approx 4,708,854 \], shows a significant rise.
Conversely, Bulgaria's population is expected to decrease, as shown by calculating \[ P = 7,148,785 \times (1 - 0.00768)^{15} \approx 6,336,505 \].
These predictions help in planning and resource allocation, ensuring that societies can adapt to population changes efficiently.
Making accurate future estimates is essential for governments and organizations as they prepare for potential challenges or opportunities linked to demographic shifts.