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:
where:
- is the population at time ,
- is the initial population,
- is the growth rate,
- 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:
where:
- is the population at time ,
- is the initial population,
- is the growth rate,
- is the time in years from the initial time.
Given
- in 1920,
- in 1930.
First, we need to find the growth rate .
- Calculate the growth rate :
Take the natural logarithm of both sides:
Now, let's calculate :
- Calculate the population in 2000 (which is 80 years after 1920):
Calculate the exponent:
Now, calculate the population:
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:
- How would the result change if the population in 1930 were different?
- Can you derive the same result using a different growth model?
- What is the population doubling time for this city based on the growth rate ?
- How does exponential growth differ from linear growth in terms of population modeling?
- How would the model change if the growth rate 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
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