A scientist used a microscope to count the number of bacterial cells in a petri dish every hour. The graph shows the data collected.


Which function most accurately represents this data, where n is the number of bacteria and t is the time elapsed, in hours?

To determine the function that most accurately represents the relationship between the number of bacteria \(n\) and the time \(t\) in hours, we need to consider the pattern in bacterial growth.

When bacteria grow, they typically follow one of these common models:

1. **Linear Growth**: If the number of bacteria increases by a fixed amount each hour, the graph would be a straight line.
   - Function form: \(n(t) = at + b\)
   - This is unlikely for bacteria, as they usually don't grow at a constant rate.

2. **Exponential Growth**: If the bacteria double (or multiply by a constant factor) at regular intervals, the graph would be an upward curve.
   - Function form: \(n(t) = n_0 \cdot e^{kt}\)
   - This is typical for bacterial populations under ideal conditions, as bacteria reproduce by binary fission (each cell divides into two).

3. **Logistic Growth**: If the bacteria initially grow exponentially but the growth slows down due to limited resources (like space or nutrients in the petri dish), the graph would have an S-shape.
   - Function form: \(n(t) = \frac{n_{\text{max}}}{1 + Ae^{-kt}}\)

### Steps to Identify the Function:
1. **Observe the Shape of the Graph**: If the graph shows a steep upward curve at the beginning and then levels off over time, it suggests logistic growth. If the curve is consistently rising without leveling off, it suggests exponential growth.

2. **Match the Data**: Depending on whether the number of bacteria is increasing exponentially or approaching a carrying capacity (which would indicate logistic growth), the appropriate function can be selected.

### Conclusion:
- **Exponential Growth** is most likely if the bacteria keep increasing rapidly, without signs of leveling off, and the function is of the form \(n(t) = n_0 \cdot e^{kt}\).
- **Logistic Growth** is likely if the bacteria initially increase rapidly but slow down as they approach a maximum population, giving the function \(n(t) = \frac{n_{\text{max}}}{1 + Ae^{-kt}}\).

Would you like more details about how to find the parameters of the equation based on the graph?

Here are 5 follow-up questions to explore further:
1. How can you distinguish between exponential and logistic growth on a graph?
2. What conditions would cause bacteria to follow a logistic growth model instead of exponential growth?
3. How do we determine the constants \(k\) and \(n_0\) in an exponential model?
4. What happens to the bacterial population when resources in the petri dish become scarce?
5. Can bacterial growth ever be linear? Under what circumstances?

**Tip**: Always check for signs of "carrying capacity" in biological systems, as this often indicates logistic growth.

A scientist used a microscope to count the number of bacterial cells in a petri dish every hour. The graph shows the data collected. Which function most accurately represents this data, where n is the number of bacteria and t is the time elapsed, in hours?

This page helps you identify the most accurate function to represent bacterial growth based on data collected every hour. It explains exponential and logistic growth models with examples of bacterial reproduction, including formulas and graph interpretation for Grades 9-12.

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