The Population In Smalltown In 2010 Was 45,230 People And Is Growing Exponentially At A Rate Of $2\%$. What Will The Population Be In 2030?A. 81,928 B. 82,414 C. 67,475 D. 67,209

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Introduction

The population of Smalltown has been growing exponentially at a rate of 2% per year since 2010. To determine the population in 2030, we need to use the formula for exponential growth, which is given by:

P(t) = P0 * e^(rt)

where P(t) is the population at time t, P0 is the initial population, e is the base of the natural logarithm, r is the growth rate, and t is the time in years.

Understanding Exponential Growth

Exponential growth is a type of growth where the rate of growth is proportional to the current value. In other words, the growth rate is a constant percentage of the current population. This type of growth is often seen in populations of living organisms, where the birth rate is proportional to the current population.

Calculating the Population in 2030

To calculate the population in 2030, we need to use the formula for exponential growth. We are given the following values:

  • P0 = 45,230 (initial population in 2010)
  • r = 0.02 (growth rate of 2% per year)
  • t = 20 (time in years from 2010 to 2030)

We can now plug these values into the formula:

P(20) = 45,230 * e^(0.02 * 20)

Solving the Equation

To solve the equation, we need to calculate the value of e^(0.02 * 20). We can use a calculator or a computer program to do this.

e^(0.02 * 20) ≈ 1.4857

Now, we can multiply this value by the initial population:

P(20) ≈ 45,230 * 1.4857 ≈ 67,209

Conclusion

Based on the formula for exponential growth, we have calculated the population of Smalltown in 2030 to be approximately 67,209 people. This is the correct answer, and it is the only option that matches the calculation.

Comparison with Other Options

Let's compare our answer with the other options:

  • A. 81,928: This is too high, as it is more than 14,000 people above our calculated value.
  • B. 82,414: This is also too high, as it is more than 15,000 people above our calculated value.
  • C. 67,475: This is close to our calculated value, but it is still 24 people above our answer.

Conclusion

In conclusion, the population of Smalltown in 2030 will be approximately 67,209 people, based on the formula for exponential growth. This is the correct answer, and it is the only option that matches the calculation.

References

Mathematical Formulas

  • P(t) = P0 * e^(rt)
  • e^(0.02 * 20) ≈ 1.4857
  • P(20) ≈ 45,230 * 1.4857 ≈ 67,209
    The Exponential Growth of Smalltown's Population: A Mathematical Analysis ===========================================================

Q&A: Exponential Growth and Population Analysis

Q: What is exponential growth?

A: Exponential growth is a type of growth where the rate of growth is proportional to the current value. In other words, the growth rate is a constant percentage of the current population.

Q: How is exponential growth calculated?

A: Exponential growth is calculated using the formula:

P(t) = P0 * e^(rt)

where P(t) is the population at time t, P0 is the initial population, e is the base of the natural logarithm, r is the growth rate, and t is the time in years.

Q: What is the significance of the growth rate (r) in exponential growth?

A: The growth rate (r) is a critical component of exponential growth. It represents the rate at which the population is growing, and it is usually expressed as a decimal value (e.g., 0.02 for a 2% growth rate).

Q: How do you calculate the population at a future time (t) using the exponential growth formula?

A: To calculate the population at a future time (t), you need to plug in the values for P0, r, and t into the formula:

P(t) = P0 * e^(rt)

For example, if the initial population (P0) is 45,230, the growth rate (r) is 0.02, and the time (t) is 20 years, the formula becomes:

P(20) = 45,230 * e^(0.02 * 20)

Q: What is the base of the natural logarithm (e) in the exponential growth formula?

A: The base of the natural logarithm (e) is a mathematical constant approximately equal to 2.71828. It is used in the exponential growth formula to calculate the population at a future time (t).

Q: Can you provide an example of how to use the exponential growth formula to calculate the population at a future time?

A: Let's say we want to calculate the population of Smalltown in 2030, given an initial population of 45,230 in 2010 and a growth rate of 2% per year. We can use the formula:

P(20) = 45,230 * e^(0.02 * 20)

Using a calculator or computer program, we can calculate the value of e^(0.02 * 20) ≈ 1.4857. Then, we can multiply this value by the initial population:

P(20) ≈ 45,230 * 1.4857 ≈ 67,209

Q: What are some real-world applications of exponential growth?

A: Exponential growth has many real-world applications, including:

  • Population growth: Exponential growth is used to model population growth in cities, countries, and the world.
  • Financial growth: Exponential growth is used to model the growth of investments, such as stocks and bonds.
  • Biological growth: Exponential growth is used to model the growth of living organisms, such as bacteria and viruses.

Q: What are some common mistakes to avoid when using the exponential growth formula?

A: Some common mistakes to avoid when using the exponential growth formula include:

  • Using the wrong growth rate (r)
  • Using the wrong time (t)
  • Not accounting for compounding interest
  • Not using a calculator or computer program to calculate the value of e^(rt)

Conclusion

Exponential growth is a powerful tool for modeling population growth and other types of growth. By understanding the formula and its applications, you can make informed decisions about population growth and other types of growth.