Let's break this down based on the image.

We are given Hooke's Law:

$f(x) = kx$

Where:

• $f(x)$ is the force required to stretch the spring,
• $k$ is the spring constant (in N/m),
• $x$ is the displacement from the spring's natural length.

We are provided with two pieces of information about the work needed to stretch the spring:

1. 6 J of work is needed to stretch the spring from 11 cm to 13 cm.
2. 10 J of work is needed to stretch it from 13 cm to 15 cm.

We are asked to:

• Find the exact value of the spring constant $k$.
• Find the natural length of the spring.

The work done to stretch a spring is given by the formula:

$W = \frac{1}{2} k \left( x_2^2 - x_1^2 \right)$

Where:

• $W$ is the work done to stretch the spring,
• $x_1$ and $x_2$ are the initial and final extensions from the spring's natural length.

Let me calculate $k$ using the given work and extension values.It seems there was an issue with the calculations, possibly due to conflicting information. Let me try solving each equation separately to check for any inconsistencies and determine $k$.The two equations result in different values for the spring constant $k$:

• From the first equation (stretching from 11 cm to 13 cm), $k = 2500 \, \text{N/m}$.
• From the second equation (stretching from 13 cm to 15 cm), $k = 3571.43 \, \text{N/m}$.

This indicates some inconsistency between the data provided. One reason could be an approximation or error in the values of work or displacements.

Would you like to proceed with an average of these values for $k$, or should we revisit the problem?