06. Modelo poblacional, ¿cuánta población habrá en el poblado en 2025?

Key Concepts

• Modelo de población (Population model)
• Tasas de natalidad y mortalidad (Birth and mortality rates)
• Tiempo (t) medido en años
• Población inicial (P sub 0)
• Constante de proporcionalidad (k)
• Exponencial (e)
• Logaritmo natural (ln)
• Estimación de la población (Population estimation)
• Redondeo (Rounding)

Main Topics and Key Points

• Problem Statement: Estimating the population of a town in 2025, given that it had 5000 inhabitants in 2000 and 10,000 in 2017, assuming constant birth and mortality rates.
• Model Selection: The problem uses a simple population model based on the assumption of constant birth and mortality rates. The model is: P(t) = P_0 * e^(kt) where:
  • P(t) is the population at time t.
  • P_0 is the initial population.
  • e is the base of the natural logarithm.
  • k is a constant related to the growth rate.
  • t is the time in years.
• Defining Time: The year 2000 is set as t = 0.
• Initial Population: In the year 2000, the population was 5000, so P(0) = P_0 = 5000.
• Population in 2017: In 2017, the population was 10,000. Since 2017 is 17 years after 2000, P(17) = 10000.
• Finding the Constant k:
  • Substitute t = 17 and P(17) = 10000 into the population model: 10000 = 5000 * e^(17k).
  • Solve for k:
    • Divide both sides by 5000: 2 = e^(17k).
    • Take the natural logarithm of both sides: ln(2) = 17k.
    • Divide by 17: k = ln(2) / 17.
    • Calculate k: k ≈ 0.04077.
• Population Model with Found Constants: The population model becomes P(t) = 5000 * e^(0.04077t).
• Estimating Population in 2025:
  • Determine the value of t for 2025. Since 2025 is 25 years after 2000, t = 25.
  • Substitute t = 25 into the population model: P(25) = 5000 * e^(0.04077 * 25).
  • Calculate P(25):
    • 0. 04077 * 25 ≈ 1.01925.
    • e^(1.01925) ≈ 2.7711.
    • 5000 * 2.7711 ≈ 13855.5.
• Rounding the Result:
  • Initially, the result is rounded to the nearest whole number: 13,856 inhabitants.
  • Then, it's argued that due to the approximate nature of the model, rounding to the nearest hundred (13,900) or thousand (14,000) is more appropriate.
• Final Answer: The estimated population in 2025 is approximately 14,000 inhabitants.

Important Examples, Case Studies, or Real-World Applications Discussed

• The entire video is a real-world application of the population model to estimate the population of a town.

Step-by-Step Processes, Methodologies, or Frameworks Explained

1. Define the problem: Clearly state what needs to be estimated.
2. Choose the appropriate model: Select the population model P(t) = P_0 * e^(kt).
3. Define the time scale: Set the initial year as t = 0.
4. Identify known values: Determine P_0 and P(t) at a specific time t.
5. Solve for the constant k: Substitute the known values into the model and solve for k.
6. Create the specific model: Substitute P_0 and k into the general model.
7. Estimate the future population: Substitute the desired time t into the specific model and calculate P(t).
8. Round the result: Round the result to an appropriate level of precision based on the accuracy of the model.

Key Arguments or Perspectives Presented, with Their Supporting Evidence

• The importance of rounding: The video argues that it's misleading to present the population estimate with high precision (e.g., 13,856) because the model is based on assumptions and approximations. Rounding to the nearest hundred or thousand is more realistic and reflects the uncertainty in the estimate.

Notable Quotes or Significant Statements with Proper Attribution

• "En cierto poblado había 5000 habitantes en el año 2000 si en el año 2017 se alcanzó una población de 10.000 habitantes estimé la población que habrá en el año 2025 suponiendo que las tasas de natalidad y mortalidad se mantienen constantes" (Problem statement).
• "Esta suposición se hace nada más para poder utilizar el modelo simple de población" (Justification for the constant rates assumption).
• "Algo más correcto sería estimar esto por ejemplo a las centenas entonces esto lo redondeamos como 900 y diríamos habrá aproximadamente 13 mil 900 habitantes o incluso podríamos redondear lo a los miles" (Argument for rounding to reflect model uncertainty).

Technical Terms, Concepts, or Specialized Vocabulary with Brief Explanations

• Modelo de población (Population model): A mathematical equation that describes how a population changes over time.
• Tasas de natalidad y mortalidad (Birth and mortality rates): The rates at which births and deaths occur in a population.
• Constante de proporcionalidad (k): A constant that determines the rate of population growth in the model.
• Exponencial (e): A mathematical constant approximately equal to 2.71828, used in exponential growth models.
• Logaritmo natural (ln): The logarithm to the base e.

Logical Connections Between Different Sections and Ideas

• The problem statement leads to the selection of the population model.
• The initial population and population in 2017 are used to find the constant k.
• The calculated k is used to create a specific population model.
• The specific model is used to estimate the population in 2025.
• The estimated population is then rounded to reflect the uncertainty of the model.

Data, Research Findings, or Statistics Mentioned

• Population in 2000: 5000 inhabitants.
• Population in 2017: 10000 inhabitants.
• Estimated population in 2025: Approximately 14000 inhabitants.

Brief Synthesis/Conclusion of the Main Takeaways

The video demonstrates how to use a simple population model to estimate future population size based on historical data. It highlights the importance of understanding the assumptions of the model and rounding the results appropriately to reflect the uncertainty in the estimate. The key takeaway is that mathematical models are tools for estimation, and the results should be interpreted with caution and common sense.