Final answer:
The student's question involves the use of the exponential decay formula to calculate population changes in a small town, demonstrating how to work out population projections over several years, including the projection after 4 and 12 years in the context of an 8% annual decrease.
Step-by-step explanation:
The question pertains to exponential decay in a small town's population, which is decreasing by 8% each year. To calculate the population after several years, one would use the formula P = P0(1 - r)t, where P is the population at time t, P0 is the initial population, r is the rate of decrease (expressed as a decimal), and t is the number of years.
For example, if the initial population is 2,148 and it's decreasing at a rate of 8% annually, after 4 years the population would be calculated as follows: P = 2,148(1 - 0.08)4. To solve this, we first calculate 1 - 0.08 = 0.92, then raise 0.92 to the power of 4, and finally multiply the result by 2,148 to get the future population.
To verify whether the population indeed approaches 7 billion in 260 years at a doubling rate every 10 years, one would use a similar exponentiation process with the initial population of 100 and apply the doubling rule for the number of required periods (260/10 in this case).