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!