Final answer:
The question pertains to verifying the population growth using exponential growth functions and involves calculations to confirm population sizes at certain times based on given growth rates. Using the provided formulas and starting populations, we can calculate the population of a town doubling every 10 years and also predict growth for a larger population with a fixed growth rate.
Step-by-step explanation:
Exponential Population Growth
The question involves an exponential growth function f(x)=12,000(1.036)^x which models the population of a town in years after 2008. To verify the population growth given a certain rate and initial population, we need to use this growth formula accordingly:
- For the population to double every 10 years, we use the formula P = P0 * 2^(t/10), where P0 is the initial population and t is the time in years.
- To apply this to the town starting with 100 residents in 1900 and doubling every 10 years, we see that by the year 2000, the population would have doubled 10 times (2000 - 1900 = 100 years; 100 / 10 = 10 doublings).
- The calculation would be 100 * 2^10, which equals 102,400, confirming the claim mentioned.
- Then, if such growth were to continue, in 260 years, the population would be 100 * 2^(260/10), which indeed gets close to 7 billion.
For a country with a 20 million population growing at a fixed rate, we use the general exponential growth formula P = P0 * e^(r * t), with 'r' representing the growth rate and 't' the time in years. If the rate is 1% per year, the population next year would be 20 million * e^(0.01).
Lastly, for a country undergoing demographic transition with a certain delay and change in birth/death rates, we must use a more complex population model that considers these rates and the time of transition to estimate the final population.