Suppose the population of a certain bacteria in a laboratory sample is 100. If it doubles in population every 6 hours, what is the growth rate? How many bacteria will there be in two days?

Let's solve this step by step.

### 1. Determine the Growth Rate:
Given that the population doubles every 6 hours, we can use the formula for exponential growth:

\[
P(t) = P_0 \times (1 + r)^t
\]

where:
- \( P(t) \) is the population at time \( t \),
- \( P_0 \) is the initial population,
- \( r \) is the growth rate per time period,
- \( t \) is the number of time periods.

Here, the population doubles every 6 hours, so after 6 hours, the population is:

\[
2P_0 = P_0 \times (1 + r)^1
\]

This simplifies to:

\[
2 = (1 + r)
\]

So, the growth rate per 6 hours is:

\[
r = 1
\]

Thus, the population grows by 100% every 6 hours.

### 2. Calculate the Population After Two Days (48 hours):
Two days is 48 hours. Since the population doubles every 6 hours, we first calculate the number of 6-hour periods in 48 hours:

\[
\text{Number of 6-hour periods} = \frac{48}{6} = 8
\]

Since the population doubles each time, we can calculate the population after 8 periods as:

\[
P(48) = P_0 \times 2^8
\]

Given \( P_0 = 100 \):

\[
P(48) = 100 \times 2^8 = 100 \times 256 = 25600
\]

### Final Answer:
The growth rate per 6 hours is 100%, and after two days (48 hours), there will be **25,600 bacteria**.

Would you like more details or have any questions?

### Related Questions:
1. What would the population be after 3 days?
2. How would the calculation change if the population tripled every 6 hours?
3. What is the formula to calculate the time required for the population to reach a certain number?
4. How can this growth model be used in other real-life scenarios?
5. What assumptions are made in this exponential growth model?

**Tip**: Exponential growth can lead to very large numbers quickly. Understanding the doubling time is crucial in predicting future population sizes.

Explore how bacteria population doubles every 6 hours in a laboratory sample. Learn the exponential growth formula and calculate the population after 48 hours. Detailed step-by-step solution provided.

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