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The problem discusses how a string stretches linearly with the mass applied. We are given the following information:

- No additional mass: the hanger is **30.6 cm** off the ground.
- With **150 grams** of mass: the hanger is **16.2 cm** off the ground.

We are asked to find the mass of a wooden block that causes the hanger to be **24.2 cm** off the ground.

### Step-by-step solution:

1. **Define the relationship:**
   Since the stretch varies linearly with mass, we can assume the position of the hanger follows the equation of a line:
   \[
   h = h_0 - k \cdot m
   \]
   where:
   - \( h \) is the height of the hanger off the ground,
   - \( h_0 \) is the initial height (30.6 cm when no mass is added),
   - \( k \) is the rate of change (slope) of the height with mass,
   - \( m \) is the mass in grams.

2. **Find the slope \( k \):**
   Using the two given points, \((0 \, \text{grams}, 30.6 \, \text{cm})\) and \((150 \, \text{grams}, 16.2 \, \text{cm})\), we can calculate the slope \( k \):
   \[
   k = \frac{30.6 - 16.2}{0 - 150} = \frac{14.4}{-150} = -0.096 \, \text{cm/gram}
   \]

3. **Write the equation:**
   Now, we have the equation of the line:
   \[
   h = 30.6 - 0.096 \cdot m
   \]
   
4. **Determine the mass of the wooden block:**
   We are given that the wooden block stretches the spring to a height of **24.2 cm**. Substitute \( h = 24.2 \) into the equation and solve for \( m \):
   \[
   24.2 = 30.6 - 0.096 \cdot m
   \]
   \[
   0.096 \cdot m = 30.6 - 24.2 = 6.4
   \]
   \[
   m = \frac{6.4}{0.096} = 66.67 \, \text{grams}
   \]

So, the mass of the wooden block is approximately **66.7 grams**.

Would you like further details or clarifications? Here are 5 questions you could explore:

1. How does the spring constant relate to the slope of the line in this problem?
2. What would happen to the equation if the mass varied non-linearly with height?
3. Can you graph the relationship between the mass and height for this spring system?
4. How would the system behave if you used a heavier block? Would the spring obey Hooke's law indefinitely?
5. Could the spring’s material impact the linearity of its stretch?

**Tip:** Linear relationships like this one can often be solved using the concept of proportionality, which simplifies many real-world physics problems.

A wooden block stretched the spring so the hanger was 24.2 cm off the ground. Determine the mass of the wooden block, to at least 1 decimal place.

In this physics problem, learn how to determine the mass of a wooden block that stretches a spring, given a linear relationship between mass and height. Explore key algebraic concepts and formulas such as h = h_0 - k * m and the slope formula. Ideal for students in Grades 9-11.

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