Problem 17 - 5 Steps to A 5: AP Calculus AB 2021

Question

The population in a city was approximately 750,000 in 1980 , and grew at a rate of \(3 \%\) per year. If the population growth followed an exponential growth model, find the city's population in the year 2002 .

Step-by-Step Solution

Answer

Answer: The population in the city in 2002 was approximately 1,173,306.

Step 1: Convert the growth rate to decimal

To convert the percentage growth rate to a decimal, divide the percentage value by 100: \(r = \frac{3}{100} = 0.03\)

Step 2: Calculate the number of years

Calculate the number of years that have elapsed from 1980 to 2002: \(t = 2002 - 1980 = 22 \text{ years}\)

Step 3: Plug values into the exponential growth formula

Now, we can plug the values we found into the exponential growth formula: \(P(t) = P_0(1 + r)^t\) \(P(22) = 750000(1 + 0.03)^{22}\)

Step 4: Calculate the population in 2002

Now, compute the population in 2002: \(P(22) = 750000(1.03)^{22} \approx 1173306\) The population in the city in 2002 was approximately 1,173,306.

Key Concepts

Population Growth

Population growth is an essential concept in understanding how a city's population changes over time. It refers to the increase in the number of people living in a particular area. Population growth can be influenced by various factors, such as birth rates, death rates, and migration.
In many cases, populations grow at a consistent rate, which can be either linear or exponential. Exponential growth is especially applicable when the growth rate is constant over time, resulting in a rapid increase as the population base becomes larger. Understanding the fundamentals of population growth helps us predict how quickly a city or region might expand over a given time frame, impacting infrastructure and resources.

Growth Rate Calculation

Calculating the growth rate is a critical step in predicting future populations. It allows us to understand how quickly a population is increasing. The growth rate is usually expressed as a percentage.
To convert this percentage into a decimal for use in calculations, simply divide it by 100. For example, if a city's population grows at a rate of 3% annually, the decimal equivalent would be 0.03.
- **To calculate growth rate in decimals:** - Use the formula: \[ r = \frac{\text{percentage rate}}{100} \]- Consistent growth rates can significantly impact future population projections, reflecting exponential growth patterns over the years.

Exponential Growth Formula

The exponential growth formula is a powerful tool in calculating the future population of a city when the growth rate is constant over the years. It captures the idea that as the population base expands, each year the increment becomes larger, reflecting exponential growth.
The formula to calculate future population is:\[ P(t) = P_0(1 + r)^t \]Where:

• \( P(t) \) : future population after time \( t \)
• \( P_0 \) : initial population at the starting point
• \( r \) : growth rate expressed as a decimal
• \( t \) : time elapsed in years

To use this formula, substitute the initial population, growth rate, and the number of years since the start point. For instance, calculating the population of a city from 1980 to 2002 with an initial population of 750,000 growing at a rate of 3% per year involves plugging these values in:
- Initial population \( P_0 = 750,000\)- Growth rate \( r = 0.03\)- Number of years \( t = 22\)The calculation would show a significant increase in population, illustrating the power of exponential growth over extended periods.