The doubling period of a bacterial population is 15 minutes. At time t = 100 minutes, the bacterial population was 60000. Find the size of the bacterial population after 5 hours. 


The problem you're dealing with involves exponential growth, where the bacterial population doubles every 15 minutes. To solve it, we'll use the exponential growth model formula:

\[
P(t) = P_0 \cdot 2^{\frac{t}{T}}
\]

Where:
- \( P(t) \) is the population at time \( t \).
- \( P_0 \) is the initial population (at \( t = 0 \)).
- \( T \) is the doubling time (in this case, 15 minutes).
- \( t \) is the time elapsed.

### Step 1: Find the initial population \( P_0 \)
At \( t = 100 \) minutes, the bacterial population was 60,000. We can use this information to find \( P_0 \).

Substitute into the equation:

\[
60000 = P_0 \cdot 2^{\frac{100}{15}}
\]

Now, simplify the exponent:

\[
\frac{100}{15} = \frac{20}{3}
\]

So, the equation becomes:

\[
60000 = P_0 \cdot 2^{\frac{20}{3}}
\]

Now, solve for \( P_0 \):

\[
P_0 = \frac{60000}{2^{\frac{20}{3}}}
\]

Using a calculator for the exponent:

\[
2^{\frac{20}{3}} \approx 9.080
\]

Thus:

\[
P_0 = \frac{60000}{9.080} \approx 6607.9
\]

So the initial population \( P_0 \) is approximately 6,608.

### Step 2: Find the population after 5 hours
Next, we calculate the population after 5 hours. Since 5 hours is 300 minutes, we plug this into the formula:

\[
P(300) = 6608 \cdot 2^{\frac{300}{15}}
\]

Simplifying the exponent:

\[
\frac{300}{15} = 20
\]

So:

\[
P(300) = 6608 \cdot 2^{20}
\]

We know that \( 2^{20} = 1,048,576 \), so:

\[
P(300) = 6608 \cdot 1,048,576 \approx 6,926,901,248
\]

Thus, the bacterial population after 5 hours will be approximately 6.93 billion.

### Would you like to discuss the solution details or ask further questions?

Here are 5 related questions for you:
1. How does the population change if the doubling period is shortened?
2. What happens to the population size if the initial population is larger?
3. Can you determine how many times the population doubles in a given time?
4. How would you calculate the population after 10 hours?
5. Can you derive a formula to predict the time when the population will reach a specific size?

**Tip:** Exponential growth models are widely used to describe processes where quantities grow rapidly over time, such as populations or investments.

The doubling period of a bacterial population is 15 minutes. At time t = 100 minutes, the bacterial population was 60000. Find the size of the bacterial population after 5 hours.

Discover how to calculate the bacterial population after 5 hours, given a doubling period of 15 minutes and an initial population of 60,000 at t = 100 minutes. Learn to apply the exponential growth model to solve real-world problems. Suitable for students in Grades 9-12.

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