A bacteria is observed undergoing exponential growth in a plate count experiment it gorws from an initial count of 22 to 31 in a period of 18 hours (calculate k from this data) what will the predicted plated count after 36hours ?

The growth of bacteria in an experiment can be modeled using the exponential growth formula:

\[
N(t) = N_0 e^{kt}
\]

where:
- \( N(t) \) is the population at time \( t \),
- \( N_0 \) is the initial population,
- \( k \) is the growth rate,
- \( t \) is time,
- \( e \) is the base of the natural logarithm (approximately 2.71828).

### Step 1: Calculate the Growth Rate \( k \)

We are given:
- Initial count \( N_0 = 22 \)
- Final count after 18 hours \( N(18) = 31 \)
- Time \( t = 18 \) hours

The equation for the population after 18 hours is:

\[
31 = 22 e^{18k}
\]

Solving for \( k \):

\[
e^{18k} = \frac{31}{22}
\]

\[
18k = \ln\left(\frac{31}{22}\right)
\]

\[
k = \frac{\ln\left(\frac{31}{22}\right)}{18}
\]

Let's calculate \( k \).

### Step 2: Calculating \( k \) 

\[
k = \frac{\ln\left(\frac{31}{22}\right)}{18}
\]The growth rate \( k \) is approximately \( 0.01905 \) per hour.

### Step 3: Predict the Plated Count After 36 Hours

To find the population after 36 hours, we use the exponential growth formula:

\[
N(36) = N_0 e^{36k}
\]

Substituting the values:

\[
N(36) = 22 \cdot e^{36 \times 0.01905}
\]

Let's calculate the predicted plated count after 36 hours.The predicted plated count after 36 hours is approximately **43.68**.

### Summary
- The growth rate \( k \) is approximately \( 0.01905 \) per hour.
- The predicted bacterial count after 36 hours is about **44** (rounded to the nearest whole number).

Would you like more details or have any questions about this solution?

### Related Questions:
1. How would the prediction change if the initial bacterial count were different?
2. How can we model bacterial growth if it does not follow exponential growth?
3. What would happen to the growth rate \( k \) if the bacteria's growth rate doubles?
4. How can we determine the doubling time of bacteria given \( k \)?
5. What are the limitations of using exponential growth models in real-life scenarios?

**Tip:** Exponential growth models are most accurate in the early stages of bacterial growth. As resources become limited, growth often slows down and follows a logistic growth pattern.

Learn how to predict bacterial growth using the exponential growth formula N(t) = N_0 e^{kt}. Solve for the growth rate k from given data and predict the bacterial count after 36 hours. Suitable for high school students studying exponential growth in biology or mathematics.

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