. A bacteria culture starts with 500 bacteria and doubles in size every half hour.
(a)[2] How many bacteria are there after 3 hours? 

The growth of the bacteria follows an exponential pattern since it doubles in size every half hour. We can express the number of bacteria at any time using the exponential growth formula:

\[
N(t) = N_0 \cdot 2^{t/\text{half-life}}
\]

where:
- \( N(t) \) is the number of bacteria at time \( t \),
- \( N_0 \) is the initial number of bacteria (500),
- \( t \) is the time in hours,
- and the half-life here is \( 0.5 \) hours (because it doubles every half hour).

### Step-by-Step Solution:

- Initial number of bacteria, \( N_0 = 500 \).
- Time elapsed, \( t = 3 \) hours.
- Half-life (doubling period) = 0.5 hours.

Substitute into the formula:

\[
N(3) = 500 \cdot 2^{3 / 0.5}
\]

Simplifying:

\[
N(3) = 500 \cdot 2^6
\]

Since \( 2^6 = 64 \):

\[
N(3) = 500 \cdot 64 = 32,000
\]

So, after 3 hours, there are **32,000 bacteria**.

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Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How would the number of bacteria change if it tripled every half hour instead of doubling?
2. How long will it take for the culture to reach 1 million bacteria?
3. What is the doubling time if the bacteria quadrupled in 1 hour?
4. How can this exponential model be altered if the rate of growth starts slowing down after a certain time?
5. What is the growth rate per hour for this bacteria culture?

**Tip:** When dealing with exponential growth, focus on the growth rate (doubling, tripling, etc.) and the time period to simplify complex scenarios.

A bacteria culture starts with 500 bacteria and doubles in size every half hour. How many bacteria are there after 3 hours?

Solve how many bacteria there are after 3 hours in a culture that starts with 500 bacteria and doubles in size every half hour. This problem uses exponential growth and the doubling time formula. Suitable for high school students studying exponential growth.

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