To determine whether the city will reach a population of 250,000 before or after 2030, we can use the given exponential model:
\[ y = 34,904.80(1.05)^x \]
where \( y \) is the population and \( x \) is the number of years since 2000. To find out when the population will reach 250,000, we can set up the equation:
\[ 250,000 = 34,904.80(1.05)^x \]
To solve for \( x \), we first divide both sides by 34,904.80:
\[ \frac{250,000}{34,904.80} = (1.05)^x \]
Calculating the left side:
\[ \frac{250,000}{34,904.80} \approx 7.15 \]
So we have:
\[ 7.15 = (1.05)^x \]
To solve for \( x \), we can take the logarithm of both sides:
\[ \log(7.15) = x \cdot \log(1.05) \]
Now, we can solve for \( x \):
\[ x = \frac{\log(7.15)}{\log(1.05)} \]
Calculating the logarithms:
- \(\log(7.15) \approx 0.8539\)
- \(\log(1.05) \approx 0.0212\)
Substituting these values into the equation gives:
\[ x \approx \frac{0.8539}{0.0212} \approx 40.19 \]
Since \( x \) represents the years since 2000, we convert this to a specific year:
\[ 2000 + 40.19 \approx 2040 \]
Now we can conclude that the population will reach 250,000 around the year 2040. Since 2040 is after 2030, the city developer will need to adhere to the old city limits map when choosing construction sites.
Thus, the correct answer is:
b. The city developer will need to use the old city limits map as the city’s population will hit 250,000 during or after 2030.