Final answer:
The predicted annual growth rate for a state's population to double by 2026 is approximately 23.33%, the monthly growth rate is 1.8%, and the continuous growth rate is around 23.1%, as calculated using the rule of 70 and adjustments for compounding periods.
Step-by-step explanation:
The student asks about calculating the annual, monthly, and continuous growth rates for a state's population that is predicted to double by the year 2026.
To calculate the growth rates, we can use the rule of 70, which is a simplified way to estimate the doubling time of an investment's growth given a fixed annual percentage increase by dividing 70 by the annual growth rate in percent.
Assuming the current year is 2023, the population will double in 3 years (2026 - 2023). Using the rule of 70:
- Annual growth rate = 70 / Doubling Time (years)
- Annual growth rate = 70 / 3 ≈ 23.33%
For the monthly growth rate, we would need to adjust this figure for monthly compounding. Assuming a monthly compounding:
- Monthly growth rate = (1 + annual growth rate)^(1/12) - 1
- Monthly growth rate ≈ (1 + 0.2333)^(1/12) - 1 ≈ 0.018 or 1.8%
To calculate the continuous growth rate, we use the natural logarithm:
- Continuous growth rate = ln(2) / Doubling Time (years)
- Continuous growth rate = ln(2) / 3 ≈ 0.231 or 23.1%
Thus, the annual growth rate is approximately 23.33%, the monthly growth rate is about 1.8%, and the continuous growth rate is roughly 23.1%.