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.