https://file.aiquickdraw.com/mathbot/file/1729796177265h2rfc2te.jpg

The question in the image asks:

"If the bacteria on the desk have a 30-minute generation period, what would you expect the population size of the bacteria to be 3 hours after cleaning the desk with bleach solution?"

We need to calculate the expected bacterial growth over 3 hours, with a known generation period of 30 minutes and an initial population of 60 bacteria.

### Step-by-step breakdown:

1. **Initial population**: 60 bacteria.
2. **Generation time**: 30 minutes. So, in 3 hours (180 minutes), the bacteria will undergo \( \frac{180}{30} = 6 \) generations.
3. **Doubling effect**: Each generation, the population doubles.

The population after 6 generations is given by the formula:
\[
P = P_0 \times 2^n
\]
where:
- \( P_0 \) is the initial population (60 bacteria),
- \( n \) is the number of generations (6 in this case).

\[
P = 60 \times 2^6 = 60 \times 64 = 3840
\]

### Answer:
The population size of the bacteria is expected to be **3,840** after 3 hours.

Let me know if you need further details or have any questions!

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Here are 5 related questions for further exploration:
1. What would the population be after 4 hours with the same generation time?
2. How would the population change if the generation time were reduced to 20 minutes?
3. If bleach only reduces the initial population by half, what would the final count be after 3 hours?
4. How does the effect of garlic extract compare to bleach based on the data in the graph?
5. Can you model the bacterial growth with an exponential growth equation?

**Tip**: Always keep track of the number of generations when calculating growth over time, especially for bacteria with consistent generation periods!

If the bacteria on the desk have a 30-minute generation period, what would you expect the population size of the bacteria to be 3 hours after cleaning the desk with bleach solution?

This problem involves calculating the bacterial growth on a desk with a 30-minute generation period, 3 hours after using bleach solution. The exponential growth formula P = P_0 * 2^n is used to determine the population size, starting with 60 bacteria and doubling every 30 minutes. This solution is suitable for high school students in Grades 9-12.

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