To adjust the population growth formula from decades to 2-year increments, first, we need to calculate the growth rate for a 2-year period. Since the growth rate is given as 15% per decade (10 years), we can convert this to a rate for 2 years using the following relationship:
The growth factor over a decade is: \[ (1 + 0.15) = 1.15 \]
To find the growth factor for a 2-year period, we can express a 10-year period as 5 two-year periods. Therefore, we need to find the 2-year growth factor \(r\) such that: \[ (1 + r)^5 = 1.15 \]
To solve for \(r\), we take the fifth root of both sides: \[ 1 + r = (1.15)^{1/5} \]
Now, we can calculate \( (1.15)^{1/5} \):
\[ (1.15)^{1/5} \approx 1.02899 \] Thus, \[ r \approx 0.02899 \]
This means the growth factor for a 2-year period is approximately 0.03 when rounded. Therefore, the formula for predicting the population based on 2-year increments is:
\[ p(d) = 1,000,000(1 + 0.03)^d \]
where \(d\) is the number of 2-year increments.
From the options provided, the correct formula for population prediction based on 2-year increments is:
3) \(p(d) = 1,000,000(1 + 0.03)^d\)