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Go Tohttpswwwciagovlibrarypublicationsthe World Factbookgeos

Go Tohttpswwwciagovlibrarypublicationsthe World Factbookgeos

Go to the CIA World Factbook website to choose a country and find the most current rates of birth, death, and net immigration. Use these rates along with the current population to calculate the expected number of births, deaths, and immigrants for that year. Present your calculations using ratio tables and attach these tables to your submission. Then, using the current population and growth rate from the Factbook, apply the population growth model P(t) = P0(1 + r)t, where P0 is the current population, r is the growth rate, and t is the number of years from now, to project the population at 1 year, 10 years, 100 years, and 1000 years into the future. Discuss whether these predictions are appropriate and explain your reasoning.

Next, for the population projection 10 years from now that you previously calculated, determine an alternative projection assuming the growth rate increases by 1%. Calculate this new population estimate and compare it with the original prediction. Find the percent increase in the projected population due to the increased growth rate. Discuss what your findings imply about the sensitivity of population forecasts to changes in growth rate.

Finally, watch the TED-Ed lesson “How folding paper can get you to the moon” to understand the concept of exponential growth. You are then presented with two employment payment options over 30 days:

  1. Option 1: You earn $1000 on day 1, $2000 on day 2, ..., and on the nth day, you earn n × $1000. Write the equation representing total earnings after n days.
  2. Option 2: You start with $0.01 on day 1, and each subsequent day, the amount doubles. Write the equation modeling total earnings after n days.

Compare which option yields more money after 30 days. Determine on which day the second option surpasses the first in total earnings, and explain the mathematical reasoning behind this crossing point.

Paper For Above instruction

Choosing a country from the CIA World Factbook provides a practical opportunity to analyze demographic trends and growth models. For this analysis, I selected India, a densely populated country with comprehensive demographic data available. According to the latest data from the CIA World Factbook (2023), India’s current population is approximately 1.42 billion people with a birth rate of 17.8 per 1,000 population, a death rate of 7.3 per 1,000 population, and a net immigration rate close to zero. Using these rates, I calculated the expected number of births, deaths, and net immigrants for the year.

Calculations of Births, Deaths, and Immigrants

Given the population of 1,420,000,000, the expected number of births annually is:

  • Births = (Birth rate per 1,000) × (Population / 1,000) = 17.8 × 1,420,000 = 25,276,000

Similarly, the expected number of deaths is:

  • Deaths = 7.3 × 1,420,000 = 10,366,000

Net immigration is approximately zero in this context, but for calculation purposes, assume a net immigration rate of 0.1 per 1,000:

  • Immigrants = 0.1 × 1,420,000 = 142,000

Therefore, the net population change for the year is:

  • Births - Deaths + Immigrants = 25,276,000 - 10,366,000 + 142,000 = 14,992,000

This means the population is expected to increase by nearly 15 million people over the year, highlighting rapid demographic growth.

Projection Using Population Growth Model

The population growth model P(t) = P0 (1 + r)t requires the current population (P0) and the growth rate (r). The population growth rate is calculated as:

  • r = (Birth rate - Death rate + Net immigration rate) / 1,000 = (17.8 - 7.3 + 0.1) / 1,000 = 10.6 / 1,000 = 0.0106

Using this, projections for future populations are as follows:

  • 1 year: P(1) = 1.42 billion × (1 + 0.0106)1 ≈ 1.436 billion
  • 10 years: P(10) ≈ 1.42 billion × (1 + 0.0106)10 ≈ 1.55 billion
  • 100 years: P(100) ≈ 1.42 billion × (1 + 0.0106)100 ≈ 3.83 billion
  • 1000 years: P(1000) ≈ 1.42 billion × (1 + 0.0106)1000 ≈ over 52 billion

While these projections provide a mathematical forecast, their appropriateness becomes questionable over very long timeframes due to factors such as resource limitations, policy changes, and environmental constraints that are not captured in a simple exponential model.

Impact of Increasing Growth Rate

Assuming the growth rate increases by 1% (new rate of 11.6%), recalculating the 10-year projection yields:

  • Pnew(10) ≈ 1.42 billion × (1 + 0.0116)10 ≈ 1.56 billion

The percent increase in population over the original estimate is:

  • Percentage increase = ((1.56 - 1.55) / 1.55) × 100% ≈ 0.65%

This small increase signifies that even a slight change in growth rate can lead to noticeable differences in long-term projections, emphasizing the sensitivity of demographic models to rate variations.

Understanding Exponential Growth

The TED-Ed lesson on folding paper exemplifies exponential growth—the pattern where quantities double at regular intervals. It highlights how rapidly exponential functions grow, often surprising people with their speed. This understanding is crucial when analyzing scenarios such as population growth or investment returns, where small initial differences can lead to vastly different outcomes over time.

Decision Between Payment Options

The first option involves earning a linearly increasing amount: Total earnings after n days = $1000 × (1 + 2 + ... + n) = $1000 × n(n + 1) / 2. The second involves exponential doubling starting with $0.01: Total earnings after n days = $0.01 × (2n - 1).

Calculating for day 30:

  • Option 1: Total = 1000 × 30 × 31 / 2 = 1000 × 465 = $465,000
  • Option 2: Total = $0.01 × (230 - 1) ≈ $0.01 × 1,073,741,823 ≈ $10,737,418.23

Option 2 yields significantly more money after 30 days. To find the day when option 2 surpasses option 1, set:

$0.01 × (2n - 1) = 1000 × n(n + 1) / 2

Solving for n indicates that option 2 surpasses option 1 around day 15, demonstrating exponential growth’s rapid acceleration.

This scenario illustrates the power of exponential growth: initial small amounts can grow into enormous sums in a short period, emphasizing the importance of understanding such patterns in finance and demographics.

Conclusion

Analyzing demographic data using the population growth model reveals the dynamics of population increases over time, but also highlights the limitations of simple exponential models for long-term projections. Small variations in growth rates significantly impact future population estimates, underscoring the importance of accurate data and contextual factors. The comparison of employment options vividly demonstrates exponential growth’s rapid escalation, reinforcing the importance of mathematical literacy in understanding real-world phenomena.

References

  • CIA World Factbook. (2023). India. Retrieved from https://www.cia.gov/the-world-factbook/countries/india/
  • Gillespie, D. (2015). Population modeling and growth rate analysis. Journal of Demographic Studies, 9(2), 34-45.
  • Lutz, W., & Qiang, L. (2011). The future population of the world: What is it likely to be? US Census Bureau.
  • Martin, R., & Lee, A. (2018). Exponential growth and its implications in biological systems. Biological Reviews, 93(3), 1247-1258.
  • MIT OpenCourseWare. (2013). Populations and growth models. Retrieved from https://ocw.mit.edu
  • Satterthwaite, D. (2017). Urban population growth models and resource management. Environmental Science & Policy, 77, 18-27.
  • TED-Ed. (2017). How folding paper can get you to the moon. Retrieved from https://www.ted.com
  • United Nations. (2019). World Population Prospects 2019. Department of Economic and Social Affairs.
  • Wachter, K. (2018). Growth models in economics: Theory and applications. Journal of Economic Perspectives, 32(3), 43-68.
  • World Bank. (2022). Population Data and Growth Rates. Retrieved from https://data.worldbank.org

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