The accompanying table shows the number of bacteria present in a certain culture over a 4 hour period, where x is the time, in hours, and y is the number of bacteria. Write an exponential regression equation for this set of data, rounding all coefficients to the nearest hundredth. Using this equation, determine the number of bacteria present after 8 hours, to the nearest whole number.
Hours (x)	Bacteria (y)
00	12571257
11	14421442
22	15151515
33	18581858
44	19641964
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To solve this, we need to perform exponential regression on the given data and then use the regression equation to predict the number of bacteria after 8 hours.

### Step 1: Arrange the data
Here’s the provided dataTo solve this, we need to perform exponential regression on the given data and then use the regression equation to predict the number of bacteria after 8 hours.

### Step 1: Arrange the data
Here’s the provided 

| Hours (x) | Bacteria (y) |
|:---------:|:------------:|
|     0     |     1257     |
|     1     |     1442     |
|     2     |     1515     |
|     3     |     1858     |
|     4     |     1964     |

### Step 2: Exponential Regression Model
An exponential regression equation has the form:

\[
y = ab^x
\]

Where:
- \( y \) is the number of bacteria,
- \( x \) is the time in hours,
- \( a \) is the initial value (number of bacteria at time \( x = 0 \)),
- \( b \) is the growth factor.

To find \( a \) and \( b \), we can perform an exponential regression using the data.

### Step 3: Using the regression equation
Once we have the regression equation, we will substitute \( x = 8 \) to find the number of bacteria after 8 hours.

Let me calculate the exponential regression and predict the number of bacteria after 8 hours.The exponential regression equation for the given data is:

\[
y = 1262.56 \cdot (1.12)^x
\]

Using this equation, the number of bacteria present after 8 hours is approximately \( 3148 \).

Would you like more details on the process or have any questions?

### Follow-up Questions:
1. How is exponential regression different from linear regression?
2. What factors could influence the growth rate in bacterial cultures?
3. How does rounding coefficients affect the accuracy of predictions?
4. What other real-world situations could be modeled using exponential growth?
5. Can exponential growth continue indefinitely? Why or why not?

**Tip:** When working with exponential growth models, always check the long-term behavior to see if a saturation point (logistic growth) is more appropriate.

The accompanying table shows the number of bacteria present in a certain culture over a 4 hour period, where x is the time, in hours, and y is the number of bacteria. Write an exponential regression equation for this set of data, rounding all coefficients to the nearest hundredth. Using this equation, determine the number of bacteria present after 8 hours, to the nearest whole number. Hours (x) Bacteria (y): 00 1257, 11 1442, 22 1515, 33 1858, 44 1964.

Learn how to perform exponential regression on a bacteria growth dataset. This problem uses the exponential growth model to predict the number of bacteria after 8 hours, rounding coefficients to the nearest hundredth. Suitable for students in Grades 10-12.

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