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The images appear to show a pulley system with a circular pulley of diameter 9.92 inches. There is a request to calculate how many rotations the pulley must complete in order for a weight to rise by a certain distance, specifically 8 inches, based on the provided text.

To find how many times the pulley rotates for the weight to rise 8 inches, we can use the relationship between the circumference of the pulley and the length of the rope displaced.

Here’s the step-by-step process:

1. **Determine the circumference of the pulley:**
   The circumference \(C\) of a circle is given by:
   \[
   C = \pi \times d
   \]
   where \(d\) is the diameter of the pulley.

2. **Calculate the number of rotations needed:**
   The pulley must rotate enough times for the rope to move a total of 8 inches. So, the number of rotations \(N\) is given by:
   \[
   N = \frac{\text{Distance to raise weight}}{\text{Circumference of pulley}}
   \]

Let's calculate this next.

### Given:
- Diameter \(d = 9.92\) inches
- Distance to raise weight = 8 inches

I'll now calculate this.The pulley must rotate approximately 0.257 times (about a quarter turn) for the weight to rise 8 inches.

Would you like further details on this calculation or have any additional questions? Here are some related questions that might interest you:

1. How would the number of rotations change if the pulley had a different diameter?
2. What formula relates the radius of a pulley to the rotations needed?
3. How does the distance moved by the weight relate to the pulley’s radius?
4. How does the rotational speed affect the time it takes for the weight to rise?
5. Can we calculate the tension in the rope based on the weight being lifted?

**Tip:** When working with rotational motion, always consider the relationship between the circumference and distance moved for accurate results.

In order for the weight to rise 8 inches, the pulley must rotate through how many rotations if the pulley diameter is 9.92 inches?

Learn how to calculate the number of rotations needed for a pulley with a diameter of 9.92 inches to lift a weight by 8 inches. This problem involves key concepts in geometry and rotational motion, ideal for students in Grades 9-12.

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