The accompanying table shows the number of bacteria present in a certain culture over a 5 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 thousandth. Using this equation, determine the number of bacteria present after 14 hours, to the nearest whole number.
Hours (x)	Bacteria (y)
00	750750
11	817817
22	911911
33	10011001
44	11131113
55	11471147

To solve this, we will first use the given data to find the exponential regression equation. The general form of an exponential regression equation is:

\[
y = a \cdot e^{bx}
\]

Where:
- \( y \) is the number of bacteria,
- \( x \) is the time in hours,
- \( a \) and \( b \) are constants we need to determine.

The data provided is:
- \( (0, 750750) \)
- \( (1, 817817) \)
- \( (2, 911911) \)
- \( (3, 10011001) \)
- \( (4, 11131113) \)
- \( (5, 11471147) \)

We'll use exponential regression techniques to find the best-fitting values for \( a \) and \( b \), and then round these values to the nearest thousandth. After that, we can use this equation to predict the number of bacteria after 14 hours.

Let me calculate that for you.The exponential regression equation for the given data is:

\[
y = 1818152.335 \cdot e^{0.398x}
\]

Using this equation, the predicted number of bacteria after 14 hours is approximately **478,100,416**.

Would you like more details, or do you have any questions?

Here are five related questions to expand your understanding:

1. What is the general form of an exponential growth model, and how is it different from a linear model?
2. How does the growth rate (exponent \( b \)) affect the shape of an exponential curve?
3. Why are logarithmic transformations sometimes used with exponential data?
4. Can exponential regression be used for all kinds of growth data, or are there limitations?
5. How does the value of \( a \) influence the behavior of the curve at \( x = 0 \)?

**Tip:** When using exponential regression, it's important to make sure that the data is best described by an exponential relationship and not another type of curve.

The accompanying table shows the number of bacteria present in a certain culture over a 5-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 thousandth. Using this equation, determine the number of bacteria present after 14 hours, to the nearest whole number.
Hours (x)  Bacteria (y)
00  750750
11  817817
22  911911
33  10011001
44  11131113
55  11471147

Learn how to solve an exponential regression problem to model the growth of bacteria over time using given data. This detailed solution walks you through finding an equation, rounding coefficients, and predicting bacterial growth after 14 hours. Suitable for high school students in Grades 10-12.

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