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The exercise involves solving questions related to population growth using a given formula, which is usually the exponential growth formula:

\[
P = P_0 \times e^{rt}
\]

where \( P \) is the final population, \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the time period.

Let's analyze the fifth question in your image:

**Question 5:** *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?*

To find the growth rate \( r \), we use the doubling time formula, since the bacteria double every 6 hours:

\[
P = P_0 \times 2^{t/d}
\]

where \( d \) is the doubling period (6 hours), \( t \) is the time in hours, and \( P_0 \) is the initial population.

Since the bacteria double every 6 hours, the formula becomes:

\[
P = 100 \times 2^{t/6}
\]

**1. Finding the growth rate \( r \):**

We know from the formula \( P = P_0 \times e^{rt} \) that after 6 hours (\( t = 6 \)) the population doubles (\( P = 200 \)), so:

\[
200 = 100 \times e^{r \times 6}
\]

We solve this for \( r \).

**2. Finding the population after two days (48 hours):**

After establishing \( r \), we can calculate the population after 48 hours using:

\[
P = 100 \times e^{r \times 48}
\]

Let's first compute the growth rate \( r \) and then use it to find the population after two days.The solution provides multiple complex values for the growth rate \( r \), but we need only the real part. The correct value for \( r \) is given by \( \log(2^{1/6}) \), which simplifies to \( \frac{\log(2)}{6} \). This represents the continuous growth rate per hour for the bacteria.

Now, let's use this growth rate to find the number of bacteria after two days (48 hours):

\[
P = 100 \times e^{r \times 48}
\]

Calculating this value:It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to explore any further concepts or have any other questions about population growth or exponential functions? Here are some related topics we can discuss:

1. How can you derive the doubling time formula from the exponential growth equation?
2. What would be the implications if the doubling time of the bacteria was shorter or longer?
3. How does exponential growth compare to logistic growth in populations?
4. Can you apply similar exponential growth formulas to financial contexts like compound interest?
5. How would the growth change if we started with a different initial population size?

Tip: Understanding the exponential growth formula and its variables can greatly enhance your ability to analyze growth scenarios in both biological and financial contexts.

Solve and substitute the given values in the formula of population growth. 5. 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?

Explore how to calculate population growth in bacteria. Solve for growth rate and determine the number of bacteria after two days with a detailed step-by-step solution using exponential growth formulas.

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