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 thousandth. Using this equation, determine the number of bacteria present after 11 hours, to the nearest whole number.
Hours (x)	Bacteria (y)
00	15361536
11	16691669
22	17581758
33	19191919
44	21412141
Answer
Attempt 1 out of 2

Regression Equation: 

Final Answer: 

To find the exponential regression equation and the number of bacteria after 11 hours, let's first break down the process:

### Step 1: Exponential Regression Equation
The general form of an exponential regression equation is:

\[
y = ab^x
\]

where:
- \( y \) is the number of bacteria,
- \( x \) is the time in hours,
- \( a \) is the initial amount (when \( x = 0 \)),
- \( b \) is the growth factor (how much the bacteria multiply per hour).

To determine the coefficients \( a \) and \( b \), we use the given data and apply exponential regression.

### Step 2: Applying Exponential Regression
Given the dataTo find the exponential regression equation and the number of bacteria after 11 hours, let's first break down the process:

### Step 1: Exponential Regression Equation
The general form of an exponential regression equation is:

\[
y = ab^x
\]

where:
- \( y \) is the number of bacteria,
- \( x \) is the time in hours,
- \( a \) is the initial amount (when \( x = 0 \)),
- \( b \) is the growth factor (how much the bacteria multiply per hour).

To determine the coefficients \( a \) and \( b \), we use the given data and apply exponential regression.

### Step 2: Applying Exponential Regression
Given the 
- At \( x = 0 \), \( y = 1536 \)
- At \( x = 1 \), \( y = 1669 \)
- At \( x = 2 \), \( y = 1758 \)
- At \( x = 3 \), \( y = 1919 \)
- At \( x = 4 \), \( y = 2141 \)

We can use a calculator or software to compute the regression coefficients. Let me compute that for you.

### Step 3: Calculating the Number of Bacteria After 11 Hours
Once we have the equation \( y = ab^x \), we will substitute \( x = 11 \) into the equation to find the number of bacteria after 11 hours.

Let me calculate this.The exponential regression equation is:

\[
y = 1521.551 \cdot (1.085)^x
\]

Using this equation, the number of bacteria after 11 hours is approximately:

\[
y(11) = 3747 \text{ bacteria} \quad (\text{rounded to the nearest whole number}).
\]

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

Here are 5 related questions for further exploration:
1. How does exponential growth compare to linear growth in biological populations?
2. What methods can be used to solve exponential equations algebraically?
3. How is the growth factor \( b \) in exponential models interpreted in real-life scenarios?
4. How does a change in the initial amount \( a \) affect the overall exponential curve?
5. How would the regression equation change if bacteria grow at a decreasing rate?

**Tip**: When fitting data, always check the model's goodness of fit to ensure that the chosen model appropriately describes the data!

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 thousandth. Using this equation, determine the number of bacteria present after 11 hours, to the nearest whole number.
Hours (x)	Bacteria (y)
0	1536
1	1669
2	1758
3	1919
4	2141

This problem demonstrates how to derive an exponential regression equation from given data on bacteria growth over time. Learn how to apply the formula y = ab^x and calculate the number of bacteria after 11 hours, with coefficients rounded to the nearest thousandth. Suitable for students in Grades 9-12.

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