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The given problem provides a table of values and asks to model the data using an exponential regression of the form \( y = ab^x \), then predict the value for \( x = 10 \).

We are given:

| \( x \)  | 2    | 3    | 4    | 5    |
|---------|-------|-------|-------|-------|
| \( f(x) \) | 5.63 | 8.44 | 12.66 | 18.98 |

1. **Step 1**: To solve this, we need to find the values of \( a \) and \( b \) in the equation \( y = ab^x \) by using the points from the table.

2. **Step 2**: Use the points to set up two equations. Let's use \( (x_1, f(x_1)) = (2, 5.63) \) and \( (x_2, f(x_2)) = (3, 8.44) \) to calculate \( b \). From the form \( y = ab^x \), we have:

   \[
   5.63 = ab^2 \quad \text{and} \quad 8.44 = ab^3
   \]

3. **Step 3**: Divide the second equation by the first to eliminate \( a \):

   \[
   \frac{8.44}{5.63} = \frac{ab^3}{ab^2} = b
   \]
   Simplifying this:
   \[
   b = \frac{8.44}{5.63} \approx 1.5
   \]

4. **Step 4**: Substitute \( b = 1.5 \) back into one of the original equations to find \( a \). Using \( 5.63 = ab^2 \):

   \[
   5.63 = a(1.5)^2 = a(2.25)
   \]
   \[
   a = \frac{5.63}{2.25} \approx 2.5
   \]

5. **Step 5**: Now that we have \( a \approx 2.5 \) and \( b \approx 1.5 \), the exponential model is:

   \[
   f(x) \approx 2.5(1.5)^x
   \]

6. **Step 6**: To predict the value for \( x = 10 \):

   \[
   f(10) \approx 2.5(1.5)^{10}
   \]
   Calculating \( (1.5)^{10} \) and then multiplying by 2.5 gives:

   \[
   f(10) \approx 144.16
   \]

So, the predicted value for \( x = 10 \) is **144.163**, which matches the third option.

Would you like more details on this process or any step in particular?

Here are some related questions to deepen your understanding:
1. How would you find \( a \) and \( b \) using different points from the table?
2. What happens if the data points do not exactly follow an exponential model?
3. How do we calculate exponential regression using a calculator or software?
4. What is the significance of the base \( b \) in the model \( y = ab^x \)?
5. How would the model change if the data followed a different pattern, such as quadratic or linear?

**Tip**: When working with exponential models, the base \( b \) determines the rate of growth or decay. If \( b > 1 \), the function models exponential growth; if \( 0 < b < 1 \), it models exponential decay.

Consider the table of values below. An exponential regression in the form y = ab^x can be used to model these data. Using this exponential model, what range value can be predicted for x = 10?

Explore how to solve an exponential regression problem with given data points and predict the value of f(x) when x = 10. This solution explains how to find the exponential model y = ab^x and predicts f(10) ≈ 144.163.

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