A small pulley with a radius of 10 centimeters inches is connected by a belt to a larger pulley with a radius of 24 centimeters inches. The larger pulley is connected to a motor that causes it to rotate counterclockwise at a rate of 75 rpm (revolutions per minute). Because the two pulleys are connected by the belt, the smaller pulley also rotates in the counterclockwise direction.  What is the linear velocity of the rim of the large pulley in inches per minute?

To find the linear velocity of the rim of the large pulley, we use the following relationship:

\[
\text{Linear Velocity} = \text{Angular Velocity} \times \text{Radius}
\]

Here, the angular velocity is given in revolutions per minute (rpm), and the radius is given in centimeters. To express the velocity in inches per minute, we first convert the radius from centimeters to inches.

### Step 1: Convert the radius to inches
Given:
- Radius of the large pulley, \( r = 24 \) centimeters.

To convert centimeters to inches, use the conversion factor:

\[
1 \text{ cm} = 0.3937 \text{ inches}
\]

Thus:

\[
r = 24 \times 0.3937 = 9.4488 \text{ inches}
\]

### Step 2: Calculate the linear velocity
The angular velocity is given as \( 75 \) revolutions per minute.

The linear velocity \( v \) is calculated as:

\[
v = \omega \times r
\]

where:
- \( \omega = 75 \) rpm (angular velocity),
- \( r = 9.4488 \) inches.

Now, since 1 revolution covers a distance equal to the circumference of the pulley, which is \( 2\pi r \) inches, we have:

\[
v = 75 \times 2\pi \times 9.4488 \text{ inches per minute}
\]

### Step 3: Compute the result
Let's compute this:

\[
v = 75 \times 2 \times 3.1416 \times 9.4488
\]
\[
v \approx 75 \times 59.3613
\]
\[
v \approx 4452.0975 \text{ inches per minute}
\]

### Final Answer
The linear velocity of the rim of the large pulley is approximately **4452.10 inches per minute**.

Would you like more details on any of the steps or have any questions?

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Here are some related questions you might be interested in:
1. What is the angular velocity of the smaller pulley?
2. How does the radius of the pulley affect the linear velocity?
3. What would happen to the linear velocity if the motor speed increased?
4. How can you calculate the linear velocity if the radii and rpm of both pulleys are unknown?
5. How would the direction of rotation change if the belt crossed between the pulleys?

**Tip:** Remember that linear velocity increases directly with both the angular velocity and the radius of the pulley.

A small pulley with a radius of 10 centimeters inches is connected by a belt to a larger pulley with a radius of 24 centimeters inches. The larger pulley is connected to a motor that causes it to rotate counterclockwise at a rate of 75 rpm (revolutions per minute). Because the two pulleys are connected by the belt, the smaller pulley also rotates in the counterclockwise direction. What is the linear velocity of the rim of the large pulley in inches per minute?

Learn how to calculate the linear velocity of a large pulley with a radius of 24 cm, rotating at 75 rpm. This step-by-step guide covers key concepts like linear and angular velocity, unit conversion from centimeters to inches, and the application of related formulas. Ideal for high school students in Grades 9-12.

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