To Study The Growth Of A Population Mathematically We Use ✓ Solved

To Study The Growth Of A Population Mathematically We Use The Concept

To study the growth of a population mathematically, we use the concept of exponential models. Generally speaking, if we want to predict the increase in the population at a certain period in time, we start by considering the current population and apply an assumed annual growth rate. For example, if the U.S. population in 2008 was 301 million and the annual growth rate was 0.9%, what would be the population in the year 2050? To solve this problem, we would use the following formula: P(1 + r)^n. In this formula, P represents the initial population we are considering, r represents the annual growth rate expressed as a decimal and n is the number of years of growth. In this example, P = 301,000,000, r = 0.9% = 0.009 (remember that you must divide by 100 to convert from a percentage to a decimal), and n = 42 (the year 2050 minus the year 2008).

Plugging these into the formula, we find: P(1 + r)^n = 301,000,000(1 + 0.009)^42. Calculating, 1 + 0.009 = 1.009, and raising to the 42nd power: (1.009)^42 ≈ 1.457. Multiplying by the initial population: 301,000,000 × 1.457 ≈ 438,557,000. Therefore, the U.S. population is predicted to be approximately 438,557,000 in the year 2050.

Let’s consider the situation where we want to find out when the population will double. Using the same example, we want to determine when the population will reach 602 million, assuming the same annual growth rate of 0.9%. Set up the equation: 602,000,000 = 301,000,000(1.009)^n. Dividing both sides by 301,000,000, we get: 2 = (1.009)^n. To solve for n, we take the logarithm of both sides:

log 2 = n log (1.009). Then, n = log 2 / log (1.009). Using a calculator, log 2 ≈ 0.3010 and log (1.009) ≈ 0.00391. Thus, n ≈ 0.3010 / 0.00391 ≈ 77.0 years. This indicates that the population will double from 301 million to 602 million in approximately 77 years, meaning around the year 2085 if starting from 2008.

Now, it is your turn: Search the Internet to determine the most recent population of your home state and, if possible, its annual growth rate. If you cannot find the growth rate, you may use 0.9% for calculation purposes. Use the exponential model to predict the population of your state 10 years from now. Then, determine how long it will take for the population to double assuming a steady growth rate. Repeat the process for your city: find its current population, estimate its population in 10 years, and find the doubling time. Discuss factors that could influence population growth or decline in your city or state, and consider whether your region is experiencing growth, decline, or stability. Finally, explain how to model a population decline assuming a negative growth rate, including an example with -0.9% annual decrease and a real-world application of exponential equations beyond population modeling.

Sample Paper For Above instruction

Population growth is a fundamental aspect of demographic studies and has critical implications for resource management, urban planning, environmental sustainability, and socio-economic development. The exponential model provides a mathematical framework to understand and predict how populations evolve over time, influenced by birth rates, death rates, migration patterns, and other socio-environmental factors.

Mathematically, exponential models describe situations where the rate of change in a quantity is proportional to the current amount. The general formula used for population projections is P(t) = P_0(1 + r)^t, where P_0 is the initial population, r is the annual growth rate expressed as a decimal, and t is the time in years. This form assumes a constant growth rate and compounding at discrete intervals, making it a practical approximation for many demographic scenarios.

An illustrative example is the U.S. population in 2008, recorded at approximately 301 million. Given an assumed annual growth rate of 0.9%, we calculate the projected population in 2050—42 years later—using the formula:

P(1 + r)^n = 301,000,000 × 1.009^42 ≈ 438,557,000.

This projection highlights how exponential growth, even at modest rates, can lead to significant increases over extended periods. Understanding such projections helps policymakers plan for infrastructure, healthcare, education, and environmental management.

Furthermore, determining the time it takes for a population to double is crucial for strategic planning. By setting 2P = P(1 + r)^n, and solving for n using logarithms, we find:

n = log 2 / log (1.009) ≈ 77 years.

Thus, at a growth rate of 0.9%, it takes approximately 77 years for the population to double, from 301 million to around 602 million.

Applying these models to local contexts involves analyzing the most recent demographic data from sources like the U.S. Census Bureau or local government agencies. For example, suppose a state's current population is 5 million with an annual growth rate of 1%. In ten years, the population would be:

P(1 + r)^n = 5,000,000 × 1.01^10 ≈ 5,505,100.

Similarly, the doubling time at this rate is roughly 69 years. Discussion of factors affecting growth includes economic opportunities, migration policies, birth rates, mortality rates, urbanization trends, and environmental constraints.

If a region experiences population decline, modeled with a negative growth rate, the same formula applies, but with r

P = 2,000,000 × (1 - 0.009)^10 ≈ 1,810,000.

This approach enables planners to forecast shrinking populations and adapt services and infrastructure accordingly.

Beyond demographic applications, exponential equations are versatile tools in finance (compound interest), physics (radioactive decay), biology (bacterial growth), and environmental science (pollutant dispersion). Their utility stems from the ability to model processes where change accelerates or decelerates in proportion to the current state, illustrating the profound importance of exponential functions across multiple disciplines.

References

• Crank, J. (1975). The Mathematics of Diffusion. Oxford University Press.
• Brock, W. A., & Durlauf, S. N. (2001). Growth empirics and reality. World Bank Economic Review, 15(2), 229-272.
• United States Census Bureau. (2023). Population Estimates and Projections. https://www.census.gov
• Haub, C. (2010). World Population Growth. Population Reference Bureau.
• Seber, G. A. F., & Wild, C. J. (2003). Nonlinear Regression. Wiley-Interscience.
• Rothman, K. J. (2012). Epidemiology: An Introduction. Oxford University Press.
• Verhulst, P.-F. (1838). Notice sur la loi que la population poursuit dans son accroissement. Correspondance mathématique et physique, 10, 113-121.
• Alonso, J. A., & Leventhal, W. (2014). Mathematical Models in Population Biology. Springer.
• Press, W. H., et al. (2007). Numerical Recipes: The Art of Scientific Computing. Cambridge University Press.
• Fisher, R. A. (1930). The Genetical Theory of Natural Selection. Oxford University Press.