https://file.aiquickdraw.com/mathbot/file/17256711005447eceol1l.jpg

To solve this problem, we'll use the concept of the gear ratio, which is based on the relationship between the radii of the two gears. The number of teeth or radii of the gears determines how many rotations one gear makes in relation to the other.

### Step 1: Determine the gear ratio.
The gear ratio is given by the ratio of the radii (or diameters) of the two gears. The formula for the gear ratio is:

\[
\text{Gear Ratio} = \frac{\text{Radius of Larger Gear}}{\text{Radius of Smaller Gear}}
\]

From the image:
- Radius of the smaller gear = 3.7 cm
- Radius of the larger gear = 6.9 cm

### Step 2: Calculate the gear ratio.
\[
\text{Gear Ratio} = \frac{6.9 \, \text{cm}}{3.7 \, \text{cm}}
\]

### Step 3: Determine the angle of rotation for the larger gear.
Given that the smaller gear rotates through an angle of 300°, we use the gear ratio to find the angle of rotation for the larger gear. The relationship is:

\[
\text{Angle of Larger Gear} = \frac{\text{Angle of Smaller Gear}}{\text{Gear Ratio}}
\]

### Step 4: Perform the calculations.
Let me calculate these values.The larger gear rotates through approximately **161 degrees** when the smaller gear rotates through 300 degrees.

Would you like any additional details or have any further questions? 

Here are some related questions you might consider:
1. How does the number of teeth on a gear affect the gear ratio?
2. What would happen if the radii of the gears were equal?
3. How would you calculate the number of rotations for the larger gear if the smaller gear made several complete rotations?
4. What is the significance of gear ratios in mechanical systems?
5. How would you convert the gear rotation into linear distance if it were part of a larger machine?

**Tip:** In mechanical systems, maintaining correct gear ratios is essential for controlling speed and torque.

Two gears are adjusted so that the smaller gear drives the larger one. If the smaller gear rotates through an angle of 300°, through how many degrees will the larger gear rotate? The smaller gear has a radius of 3.7 cm, and the larger gear has a radius of 6.9 cm.

Understand how to calculate the rotation of a larger gear when driven by a smaller gear using gear ratios. This problem involves key geometric concepts and ratios. Suitable for Grades 8-10.

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