To determine the population of a city that doubles in size every 97 years, we can use the formula for exponential growth. The population \( P \) at a given time can be expressed as:
\[ P(t) = P_0 \times 2^{(t/T)} \]
where:
- \( P_0 \) is the initial population,
- \( t \) is the time elapsed,
- \( T \) is the time it takes for the population to double (97 years in this case).
Given:
- Initial population \( P_0 = 52,000 \),
- Time elapsed \( t = 291 \) years,
- Doubling time \( T = 97 \) years.
Now, we can calculate the number of doubling periods in 291 years:
\[ \text{Number of doublings} = \frac{t}{T} = \frac{291}{97} \approx 3 \]
Since \( 291 \div 97 \) equals approximately 3.00, indicating that there will be 3 complete doubling periods in 291 years.
Now, we apply this to our population formula:
\[ P(291) = 52,000 \times 2^{3} \]
Calculating \( 2^{3} \):
\[ 2^{3} = 8 \]
Next, we can find the population after 291 years:
\[ P(291) = 52,000 \times 8 = 416,000 \]
Thus, the population of the city in 291 years will be 416,000.