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The problem describes bacterial population growth following a continuous exponential growth model with a growth rate of 6% per hour.

The formula for continuous exponential growth is:

\[
P(t) = P_0 \cdot e^{rt}
\]

Where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the growth rate (as a decimal),
- \( t \) is the time in hours,
- \( e \) is Euler's number (approximately 2.71828).

### Given dataThe problem describes bacterial population growth following a continuous exponential growth model with a growth rate of 6% per hour.

The formula for continuous exponential growth is:

\[
P(t) = P_0 \cdot e^{rt}
\]

Where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the growth rate (as a decimal),
- \( t \) is the time in hours,
- \( e \) is Euler's number (approximately 2.71828).

### Given 
- Initial population \( P_0 = 265 \) bacteria,
- Growth rate \( r = 0.06 \) (since 6% = 0.06),
- Time \( t = 4 \) hours.

Now, we will substitute these values into the formula to find the population after 4 hours:

\[
P(4) = 265 \cdot e^{0.06 \cdot 4}
\]

First, calculate the exponent:
\[
0.06 \cdot 4 = 0.24
\]

Next, calculate \( e^{0.24} \):
\[
e^{0.24} \approx 1.27125
\]

Finally, calculate the population:
\[
P(4) = 265 \cdot 1.27125 \approx 336.9
\]

So, the population after 4 hours is approximately **336.9 bacteria**.

Would you like further details or explanations on any part of this calculation?

Here are 5 related questions for deeper understanding:
1. How does the population change if the growth rate increases to 10%?
2. What happens to the population after 6 hours under the same growth model?
3. Can the exponential model be used for other biological growth processes, and if so, how?
4. What is the significance of Euler's number \( e \) in continuous growth models?
5. How would the solution change if the population were modeled using a logistic growth curve?

**Tip:** In continuous exponential growth, the population grows at a rate proportional to its current size, which means growth accelerates over time.

The number of bacteria in a certain population is predicted to increase according to a continuous exponential growth model, at a relative rate of 6% per hour. Suppose that a sample culture has an initial population of 265 bacteria. Find the population predicted after four hours, according to the model.

Learn how to predict bacterial growth using the continuous exponential growth model. Given an initial population of 265 bacteria and a growth rate of 6% per hour, we calculate the population after four hours with a detailed step-by-step solution.

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