Math Problem Statement

A particular city had a population of 28 comma 000 in 1920 and a population of 31 comma 000 in 1930. Assuming that its population continues to grow exponentially at a constant​ rate, what population will it have in 2000​? Question content area bottom Part 1 The population of the city in 2000 will be    enter your response here people. ​(Round the final answer to the nearest whole number as needed. Round all intermediate values to six decimal places as​ needed.)

Solution

To solve this problem, we'll use the exponential growth model. The formula for exponential growth is:

P(t)=P0×ertP(t) = P_0 \times e^{rt}

where:

  • P(t)P(t) is the population at time tt,
  • P0P_0 is the initial population,
  • rr is the growth rate,
  • tt is the time in years from the initial time.

Given dataTo solve this problem, we'll use the exponential growth model. The formula for exponential growth is:

P(t)=P0×ertP(t) = P_0 \times e^{rt}

where:

  • P(t)P(t) is the population at time tt,
  • P0P_0 is the initial population,
  • rr is the growth rate,
  • tt is the time in years from the initial time.

Given

  • P0=28,000P_0 = 28,000 in 1920,
  • P(10)=31,000P(10) = 31,000 in 1930.

First, we need to find the growth rate rr.

  1. Calculate the growth rate rr:

31,000=28,000×e10r31,000 = 28,000 \times e^{10r}

31,00028,000=e10r\frac{31,000}{28,000} = e^{10r}

3128=e10r\frac{31}{28} = e^{10r}

Take the natural logarithm of both sides:

ln(3128)=10r\ln\left(\frac{31}{28}\right) = 10r

r=ln(3128)10r = \frac{\ln\left(\frac{31}{28}\right)}{10}

Now, let's calculate rr:

rln(1.107142857)100.10183100.010183r \approx \frac{\ln(1.107142857)}{10} \approx \frac{0.10183}{10} \approx 0.010183

  1. Calculate the population in 2000 (which is 80 years after 1920):

P(80)=28,000×e0.010183×80P(80) = 28,000 \times e^{0.010183 \times 80}

Calculate the exponent:

0.010183×800.814640.010183 \times 80 \approx 0.81464

Now, calculate the population:

P(80)=28,000×e0.8146428,000×2.25832263,233P(80) = 28,000 \times e^{0.81464} \approx 28,000 \times 2.258322 \approx 63,233

So, the population of the city in the year 2000 will be approximately 63,233 people.

Would you like to go over the details or have any questions?

Here are five related questions to consider:

  1. How would the result change if the population in 1930 were different?
  2. Can you derive the same result using a different growth model?
  3. What is the population doubling time for this city based on the growth rate rr?
  4. How does exponential growth differ from linear growth in terms of population modeling?
  5. How would the model change if the growth rate rr varied over time?

Tip: Always round intermediate calculations carefully when working with exponential models, as small errors can significantly affect the final result.

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Math Problem Analysis

Mathematical Concepts

Exponential Growth
Population Modeling
Logarithms

Formulas

P(t) = P_0 * e^(rt)
r = ln(P(t)/P_0) / t

Theorems

Exponential Growth Theorem

Suitable Grade Level

Grades 10-12