The initial size of the culture was 500 bacteria, the doubling period is 7.5 minutes, the population after 100 minutes will be 1,843,264 bacteria, and it will reach 15,000 bacteria in approximately 27.5 minutes.
Detailed Solution and Explanation:
1. Initial Size:
The initial size (N₀) is simply the count at the first measurement (15 minutes), which is 500 bacteria.
2. Doubling Period:
We can calculate the doubling period (Td) by determining the time interval between the two measurements where the population doubled. In this case, the population doubled from 500 to 1100 between 15 and 30 minutes. Therefore:
- Td = (30 minutes - 15 minutes) / 2 = 7.5 minutes (rounded to one decimal place)
3. Population after 100 minutes:
Since the population doubles every Td, we can use the formula N₁ = N₀ * 2^(t / Td), where N₁ is the population after a specific time (t) and N₀ is the initial size. For t = 100 minutes:
- N₁₀₀ = 500 bacteria * 2^(100 minutes / 7.5 minutes) ≈ 1,843,264 bacteria (rounded to nearest bacteria)
4. Time to reach 15000:
We can again use the formula N₁ = N₀ * 2^(t / Td), but this time solving for t (time) when the population reaches 15,000 bacteria:
- 15000 bacteria = 500 bacteria * 2^(t / 7.5 minutes)
- log₂(30) = t / 7.5 minutes
- t = 7.5 minutes * log₂(30) ≈ 27.5 minutes (rounded to one decimal place)
Therefore, the population will reach 15,000 bacteria in approximately 27.5 minutes.