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Let's break down the problem step by step:

1. **Initial setup**: The student starts 6 meters away from the motion sensor.
2. **Walking towards the sensor**: The student walks toward the sensor for 4 seconds at a rate of 1 meter per second. This means after 4 seconds, the student will have covered \( 4 \, \text{m} \), reducing the distance to \( 6 \, \text{m} - 4 \, \text{m} = 2 \, \text{m} \).
3. **Walking away from the sensor**: The student then walks away for 8 seconds at a rate of 0.5 meters per second. In 8 seconds, they would have walked \( 8 \times 0.5 = 4 \, \text{m} \) away from the sensor. So, the final distance from the sensor is \( 2 \, \text{m} + 4 \, \text{m} = 6 \, \text{m} \).

Now, let’s look at the **distance-time graph**:
- At \( t = 0 \), the distance is 6 meters.
- After 4 seconds of walking toward the sensor, at \( t = 4 \), the distance is 2 meters.
- After walking away for 8 seconds, at \( t = 4 + 8 = 12 \), the distance returns to 6 meters.

So, at \( t = 10 \) seconds, the distance is still 6 meters, which corresponds to point \( (10, 6) \).

The correct answer is **c. (10, 6)**.

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Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How would the graph look if the student had walked at a constant speed both ways?
2. What would change if the student had walked faster when moving away from the sensor?
3. Can you calculate the total distance the student walked?
4. How would the graph change if the initial distance from the sensor was different?
5. What is the relationship between speed and the slope of the graph?

**Tip**: When analyzing motion problems, keeping track of direction (towards vs. away from a reference point) helps in understanding how the graph will behave.

A student is walking in a straight line in front of a motion sensor. The sensor begins collecting data when the student is 6 m away. The student walks toward the sensor for 4 s at a rate of 1 m/s. Then she walks away from the sensor for 8 s at a rate of 0.5 m/s. Which of the points is on the graph of the distance versus time graph for this walk? a. (6, 0) b. (8, 2) c. (10, 6) d. (12, 6)

Analyze a student's walk towards and away from a motion sensor in this problem. Explore how speed affects the distance-time graph, covering concepts like linear motion and rates of change. A great exercise for students in Grades 6-8.

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