Problem 40 - College Algebra with Modeling and Visualization

Question

Population The population of California was about 38 million in 2007 and increasing by \(1.6 \%\) each year. Estimate the population of California in 2012 .

Step-by-Step Solution

Answer

The estimated population of California in 2012 was approximately 41.15 million.

Step 1: Identify the Initial Population and Growth Rate

The initial population of California in 2007 was 38 million. The population increases by \(1.6\%\) each year. This percentage can be converted to a decimal for calculation purposes: \(1.6\% = 0.016\).

Step 2: Determine the Number of Years for Growth

We want to find the population in 2012. The number of years between 2007 and 2012 is: \(2012 - 2007 = 5\) years.

Step 3: Apply the Compound Interest Formula for Population Growth

We use the formula for compound growth, which is similar to compound interest: \(P = P_0 (1 + r)^t\), where \(P\) is the future population, \(P_0\) is the initial population, \(r\) is the growth rate (as a decimal), and \(t\) is the number of years. Substitute the values: \(P = 38(1 + 0.016)^5\).

Step 4: Calculate the Future Population

Calculate \((1 + 0.016)^5\) first: \[(1 + 0.016)^5 = 1.083\]Now, multiply this result by the initial population:\[P = 38 \times 1.083 \approx 41.154\]Thus, the estimated population in 2012 is approximately 41.15 million.

Key Concepts

Population Modeling

Population modeling refers to the mathematical representation of population growth over time. It involves using equations to predict future changes based on current data and trends. This concept is crucial in understanding how populations grow or decline and how they are affected by various factors such as birth rates, death rates, and migration. In the exercise provided, we are dealing with a straightforward example of exponential growth, where the population is assumed to increase steadily by a fixed percentage each year.

Some key points when dealing with population modeling include:

• Identifying the initial size of the population: This is the starting point from which all future calculations will be based.
• Determining the growth rate: In our case, this is given as a percentage, which needs to be converted into a decimal for calculations.
• Considering the time period over which growth is calculated: This is critical because it directly affects the forecasted outcome.

Population models help researchers and policymakers make informed decisions regarding planning and resource allocation, based on predicted changes in population size.

Compound Interest Formula

The compound interest formula is a core mathematical tool used not only in finance but also in various fields such as population and ecological modeling. The formula is:\[ P = P_0 (1 + r)^t \] where:

• \( P \) represents the future value or size of the population or investment.
• \( P_0 \) denotes the initial population or principal amount.
• \( r \) is the growth rate or interest rate, expressed as a decimal.
• \( t \) refers to the number of time periods over which growth occurs.

This formula helps estimate how an initial quantity grows over time, taking into account compounding – where growth in one period contributes to further growth in the subsequent periods. It’s highly effective for modeling scenarios where consistent growth is expected.

Using the compound interest formula allows us to simplify the task of forecasting given uniform growth rates. In the original exercise, this formula is used to estimate California's 2012 population, showing how population models can mirror financial growth principles.

Annual Growth Rate

The annual growth rate presents the percentage increase in a given sector over a year. It’s pivotal in exponential growth contexts to express the rate of change clearly and understandably. In the context of the exercise, Californian population data shows a growth of 1.6% annually.

To use the growth rate in calculations, generally:

• Convert the percentage to a decimal (e.g., \(1.6\%\) becomes \(0.016\)).
• Account for the growth rate in the formula to find future values.

The annual growth rate helps compare and contrast growth across different periods, geographical regions, or economic situations.

It's essential to recognize that while a 1.6% growth rate may seem small yearly, compounded over multiple years, it leads to a significant increase. Understanding and calculating the annual growth rate accurately is crucial for making realistic projections in population studies and financial forecasts.