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 9 hours, to the nearest whole number.
Hours (x)	Bacteria (y)
0	1221
1	1332
2	1437
3	1576
4	1698


To find the exponential regression equation for the given data set, we assume an equation of the form:

\[
y = ab^x
\]

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

### Given Data Points:
| Hours (x) | Bacteria (y) |
|-----------|---------------|
| 0         | 1221          |
| 1         | 1332          |
| 2         | 1437          |
| 3         | 1576          |
| 4         | 1698          |

We will use exponential regression to find the best-fit values for \( a \) and \( b \). Let me calculate that for you.

The exponential regression equation for the given data set is:

\[
y = 1223.15 \times (1.09)^x
\]

Using this equation, the estimated number of bacteria after 9 hours is approximately:

\[
2573 \text{ bacteria} 
\]

Would you like further clarification or details about this solution?

Here are some related questions:
1. How would the growth factor change if the rate of bacteria growth increases?
2. Can exponential growth continue indefinitely? Why or why not?
3. What are some real-world scenarios where exponential regression is used?
4. How does exponential growth differ from linear growth?
5. What is the significance of the base \(b\) in the exponential equation?

**Tip:** In exponential growth, the quantity increases by a fixed percentage (or factor) per time unit, unlike in linear growth where the increase is by a fixed amount.

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 9 hours, to the nearest whole number. Hours (x) Bacteria (y) 0 1221 1 1332 2 1437 3 1576 4 1698

Learn how to find the exponential regression equation for bacteria growth over 4 hours, using the data provided. This solution includes detailed steps to calculate the equation and estimate the number of bacteria after 9 hours. Suitable for high school students.

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