bacteria culture starts with 20 bacteria and doubles in size every half hour. CHow many bacteria are there after 3 hours?

To solve this problem, we can use the formula for exponential growth:

\[
N(t) = N_0 \times 2^{\frac{t}{T}}
\]

Where:
- \( N(t) \) is the number of bacteria at time \( t \).
- \( N_0 \) is the initial number of bacteria.
- \( T \) is the doubling time.
- \( t \) is the elapsed time.

Given:
- \( N_0 = 20 \) (initial number of bacteria)
- \( T = 0.5 \) hours (doubling time)
- \( t = 3 \) hours (total time elapsed)

Now, substitute the values into the formula:

\[
N(3) = 20 \times 2^{\frac{3}{0.5}}
\]

Simplify the exponent:

\[
\frac{3}{0.5} = 6
\]

So, the formula becomes:

\[
N(3) = 20 \times 2^6
\]

Calculate \( 2^6 \):

\[
2^6 = 64
\]

Finally, multiply by the initial number of bacteria:

\[
N(3) = 20 \times 64 = 1280
\]

So, there will be **1,280 bacteria** after 3 hours.

Would you like more details or have any questions?

Here are five related questions to explore further:

1. How many bacteria will there be after 4 hours?
2. What if the initial number of bacteria was 50 instead of 20?
3. How long will it take for the bacteria population to reach 10,000?
4. How would the calculation change if the bacteria doubled every hour instead of every half hour?
5. How can the general formula be derived for any doubling time?

**Tip:** Understanding exponential growth is crucial because it appears in many real-world situations, such as population growth, finance, and natural processes.

Learn how to calculate bacteria growth starting with 20 bacteria, doubling every half hour. Understand the exponential growth formula and solve the problem step-by-step. After 3 hours, there will be 1,280 bacteria.

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