To convert the population growth formula from a decade basis to a 2-year basis, first, we need to determine the growth rate for every 2 years.
The original growth rate is 15% per decade (10 years). This can be expressed as a growth factor of \(1 + 0.15 = 1.15\).
To find the equivalent growth rate for each 2-year period, we use the formula for compound growth:
\[ (1 + r)^{n} = 1.15 \quad \text{where } n = \frac{10}{2} = 5 \text{ (because there are 5 increments of 2 years in 10 years)} \]
We need to find \(r\) such that:
\[ (1 + r)^{5} = 1.15 \]
To find \(1 + r\), we can take both sides to the power of \(1/5\):
\[ 1 + r = (1.15)^{\frac{1}{5}} \]
Calculating \( (1.15)^{\frac{1}{5}} \):
\[ 1 + r \approx 1.02899 \]
Thus,
\[ r \approx 0.02899 \]
This means that the growth rate every 2 years is approximately 0.02899, or about 2.9%.
Therefore, the formula for the population in terms of \(d\) (where \(d\) is the number of 2-year increments) would be:
\[ p(d) = 1,000,000(1 + 0.029)d \]
Out of the options provided, none match directly to \(p(d) = 1,000,000(1 + 0.029)^d\), but the closest choice would be approximately:
\[ p(d) = 1,000,000(1 + 0.03)^d \]
Thus, the correct response is:
p(d) = 1,000,000(1 + 0.03)^d