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We are asked to show that the value of \( k \) in the linear model \( L = km + 32 \) is 4.6, where \( L \) is the length of the spring and \( m \) is the mass attached.

The data provided are:
- \( m \) values (mass in kg): \( 2, 5, 8, 11, 14, 17 \)
- \( L \) values (length in cm): \( 41.2, 55.0, 68.8, 82.6, 96.4, 110.2 \)

The linear model \( L = km + 32 \) indicates that \( L \) is proportional to \( m \), with a slope \( k \) and an intercept of 32.

### To find \( k \):
We know that the equation \( L = km + 32 \) can be rearranged to solve for \( k \):
\[
k = \frac{L - 32}{m}
\]
We'll calculate \( k \) for different pairs of \( (m, L) \) values.

Let's compute \( k \) for the first and last points to check the consistency of \( k = 4.6 \). I'll start by calculating these values.The calculated values of \( k \) for each pair of \( (m, L) \) are approximately:

- \( k = 4.6 \) for all pairs.

This confirms that the value of \( k \) is consistently around 4.6, showing that the linear model \( L = 4.6m + 32 \) accurately describes the relationship between the mass and the length of the spring.

Would you like a detailed explanation or further clarification on any of the steps?

Here are 5 related questions:
1. How would the model change if the intercept was different?
2. What does the slope \( k = 4.6 \) represent in this context?
3. How can you use this model to predict the length \( L \) for other masses \( m \)?
4. How would you find the value of \( k \) if the relationship wasn't linear?
5. What physical principles could explain why the length depends linearly on mass?

**Tip**: When working with linear models, interpreting the slope and intercept in real-world contexts can help in understanding the system being modeled.

Items with different masses (m kg) are attached to a spring. The length of the spring (L cm) is measured for each item. The task is to show that the value of k in the linear model L = km + 32 is 4.6.

Learn how to calculate the value of k in a linear model representing the length of a spring as a function of attached mass. The problem involves using the equation L = km + 32 with mass values and spring lengths provided. Suitable for high school students studying linear relationships.

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