Problem 48
Question
In 2004 the population in Morganton, Georgia, was \(43,000 .\) The population in Morganton doubled by \(2010 .\) If the growth rate remains the same, what is the expected population in Morganton in \(2020 ?\)
Step-by-Step Solution
Verified Answer
The expected population in 2020 is approximately 181,000.
1Step 1: Understanding Doubling Time
The problem states that the population doubled from 43,000 in 2004 to a larger amount by 2010. This implies a doubling time of 6 years.
2Step 2: Calculate Growth Rate per Year
To find the growth rate per year, we use the formula for exponential growth models: \(P = P_0 (1 + r)^t\), where \(P_0\) is the initial population, \(P\) is the final population, \(r\) is the growth rate, and \(t\) is time. We know \(P\) = 86,000, \(P_0\) = 43,000, and \(t\) = 6 years (from 2004 to 2010). Plugging these values in, we get \(86,000 = 43,000(1+r)^6\).
3Step 3: Solve for Growth Rate
First, divide both sides by 43,000:\((1+r)^6 = \frac{86,000}{43,000} = 2\).To solve for \(r\), take the sixth root of both sides:\(1 + r = 2^{1/6}\).Hence, \(r = 2^{1/6} - 1\). Using a calculator for \(2^{1/6}\), the estimate for \(r\) is approximately 0.12246.
4Step 4: Predict Population in 2020
Using the growth rate found, apply it to the next 10 years from 2010 to 2020. Set up the equation:\(P = 86,000 (1 + 0.12246)^{10}\).Substitute the growth rate and calculate:\(P \approx 86,000 \times 2.1067\), resulting in approximately 181,000.
Key Concepts
Understanding Doubling TimeGrowth Rate CalculationMaking a Population Prediction
Understanding Doubling Time
When it comes to understanding exponential growth, the concept of **doubling time** is a crucial tool. It tells us how long it takes for a quantity to double at a constant growth rate. In our scenario with Morganton, Georgia, the population went from 43,000 in 2004 to 86,000 in 2010. This means the population doubled over a period of 6 years.
To determine the doubling time, it's important to recognize **this period serves as a key characteristic of exponential growth**. It provides a simple way to convey how quickly something grows. For anyone analyzing growth patterns, knowing the doubling time is akin to having a shorthand guide to the rate of growth at play.
Moreover, the doubling time can be intuitively understood as the period required for exponential growth to reach a certain threshold twice its starting point. This makes it useful for discussions around population growth, investments, and even in biology.
To determine the doubling time, it's important to recognize **this period serves as a key characteristic of exponential growth**. It provides a simple way to convey how quickly something grows. For anyone analyzing growth patterns, knowing the doubling time is akin to having a shorthand guide to the rate of growth at play.
Moreover, the doubling time can be intuitively understood as the period required for exponential growth to reach a certain threshold twice its starting point. This makes it useful for discussions around population growth, investments, and even in biology.
Growth Rate Calculation
To make future predictions, understanding how to calculate the growth rate is integral. It starts with the fundamental formula for exponential growth: \[ P = P_0 (1 + r)^t \] In which:
This calculated growth rate gives a quantifiable measure of the population increase percentage per annum, providing a foundation for subsequent predictions.
- \( P_0 \) is the initial population.
- \( P \) is the future population prediction.
- \( r \) is the growth rate per period.
- \( t \) is the time period in question.
This calculated growth rate gives a quantifiable measure of the population increase percentage per annum, providing a foundation for subsequent predictions.
Making a Population Prediction
Armed with the growth rate, we can reliably predict future population sizes using the same exponential growth formula. For Morganton’s case, we want to forecast the population for 2020, which is ten years after 2010.
With \( P = 86,000 (1 + 0.12246)^{10} \), we use the previously calculated growth rate over the 10-year span. Evaluating this gives:\[ P \approx 86,000 \times 2.1067 \]Thus, **the expected population by 2020 is around 181,000**. Using this approach allows one to make informed predictions about future conditions based on established patterns.
Population prediction through this method assists in effective planning and resource allocation. It's worth noting that certain factors such as changing birth rates, migration, and policy changes might influence future growth.
Nonetheless, this approach provides a robust starting point for thinking about future growth within communities or other scenarios.
With \( P = 86,000 (1 + 0.12246)^{10} \), we use the previously calculated growth rate over the 10-year span. Evaluating this gives:\[ P \approx 86,000 \times 2.1067 \]Thus, **the expected population by 2020 is around 181,000**. Using this approach allows one to make informed predictions about future conditions based on established patterns.
Population prediction through this method assists in effective planning and resource allocation. It's worth noting that certain factors such as changing birth rates, migration, and policy changes might influence future growth.
Nonetheless, this approach provides a robust starting point for thinking about future growth within communities or other scenarios.
Other exercises in this chapter
Problem 48
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In Exercises 49 and 50 , refer to the logistic model \(f(t)=\frac{a}{1+c e^{-k t}},\) where \(a\) is the carrying capacity. As \(c\) increases, does the model r
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