To find out when the population will hit 70,000, we need to solve the equation \( p(n) = 70 \), since the function is given in thousands. Therefore, we can set up the equation:
\[ 36.5(1.06)^x = 70 \]
Now, we divide both sides by 36.5:
\[ (1.06)^x = \frac{70}{36.5} \]
Calculating the right side:
\[ \frac{70}{36.5} \approx 1.9178 \]
So we have:
\[ (1.06)^x = 1.9178 \]
Next, we take the natural logarithm of both sides:
\[ \ln((1.06)^x) = \ln(1.9178) \]
Using the power rule of logarithms:
\[ x \ln(1.06) = \ln(1.9178) \]
Now solve for \( x \):
\[ x = \frac{\ln(1.9178)}{\ln(1.06)} \]
Now, let's calculate \( \ln(1.9178) \) and \( \ln(1.06) \):
- \( \ln(1.9178) \approx 0.69315 \)
- \( \ln(1.06) \approx 0.05827 \)
Thus,
\[ x \approx \frac{0.69315}{0.05827} \approx 11.89 \]
This means that it will take approximately 11.89 years from the year 2000. Since we round to the nearest whole number, we round \( 11.89 \) to \( 12 \).
Thus, adding 12 years to 2000 gives us:
\[ 2000 + 12 = 2012 \]
Therefore, the population will hit 70,000 in the year 2012.