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To solve the problems in this task, let's break down each part of the question.

### Task a: Determine \( a \) and \( b \) using Power Regression

The relationship between the pendulum length and the oscillation time is modeled by the power function:
\[
f(x) = b \cdot x^a
\]
where:
- \( x \) is the pendulum length in meters (we need to convert cm to meters).
- \( f(x) \) is the swing time in seconds.

#### Steps for Power Regression
1. **Convert lengths from centimeters to meters**:
   - The given lengths are in cm, so divide each by 100 to convert to meters.

2. **Perform power regression**:
   - Using the converted values, we would use a power regression technique to find the values of \( a \) and \( b \) that best fit the data.
   
Since I can handle the computation, I will proceed with these steps using the data provided.

### Task b: Calculate the Percentage Increase in Swing Time with a 40% Increase in Length

Once we have the power function \( f(x) = b \cdot x^a \):
1. Let the original pendulum length be \( x \).
2. A 40% increase in \( x \) gives a new length of \( 1.4x \).
3. Compute the new swing time using \( f(1.4x) \) and compare it to \( f(x) \).
4. The percentage increase in swing time is calculated by:
   \[
   \text{Percentage Increase} = \frac{f(1.4x) - f(x)}{f(x)} \times 100\%
   \]

I'll proceed with the calculations now.The results for each part of the question are as follows:

1. **For Task a**:
   - The parameters found through power regression are:
     - \( b \approx 2.013 \)
     - \( a \approx 0.511 \)

   So, the function describing the swing time \( f(x) \) is approximately:
   \[
   f(x) = 2.013 \cdot x^{0.511}
   \]

2. **For Task b**:
   - When the pendulum's length increases by 40%, the swing time increases by approximately **18.77%**.

Would you like a more detailed explanation or have further questions about these calculations?

Here are five related questions to expand your understanding:
1. How does the exponent \( a \) relate to the square root relationship often observed in pendulum motion?
2. What would happen to the swing time if the pendulum length doubled?
3. How would we calculate the swing time if we wanted a specific period (e.g., 2 seconds)?
4. How can we verify the accuracy of the regression model for other pendulum lengths?
5. What are some potential sources of error in experimentally determining pendulum swing times?

**Tip:** For more precise predictions, try increasing the number of data points in your regression analysis.

A pendulum is hung from a string, allowing it to swing back and forth. In an experiment, the oscillation time for 8 different lengths of the pendulum was measured. The results are shown in a table. Given that the oscillation time function can be described by the power function f(x) = b * x^a, determine the values of a and b using power regression. Additionally, calculate the percentage increase in oscillation time if the pendulum length is increased by 40%.

Learn how to determine the power regression parameters for a pendulum’s oscillation time based on its length. This problem covers power regression analysis and calculating the percentage increase in oscillation time for a 40% increase in length, suitable for students in Grades 10-12.

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