Questions: Two gears are adjusted so that the smaller gear drives the larger one, as shown in the figure. If the smaller gear rotates through an angle of 270°, through how many degrees will the larger gear rotate? The larger gear rotates through approximately 0°.

Two gears are adjusted so that the smaller gear drives the larger one, as shown in the figure. If the smaller gear rotates through an angle of 270°, through how many degrees will the larger gear rotate? The larger gear rotates through approximately 0°.
Transcript text: Two gears are adjusted so that the smaller gear drives the larger one, as shown in the figure. If the smaller gear rotates through an angle of $270^{\circ}$, through how many degrees will the larger gear rotate? The larger gear rotates through approximately $\square$ $0^{\circ}$.
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Solution

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Solution Steps

Step 1: Identify the given information
  • The smaller gear has a radius of 3.5 cm.
  • The larger gear has a radius of 6.9 cm.
  • The smaller gear rotates through an angle of 270°.
Step 2: Determine the gear ratio

The gear ratio is determined by the ratio of the radii of the two gears. \[ \text{Gear Ratio} = \frac{\text{Radius of Larger Gear}}{\text{Radius of Smaller Gear}} = \frac{6.9 \text{ cm}}{3.5 \text{ cm}} \]

Step 3: Calculate the gear ratio

\[ \text{Gear Ratio} = \frac{6.9}{3.5} \approx 1.971 \]

Step 4: Calculate the rotation of the larger gear

The angle through which the larger gear rotates is the angle of the smaller gear divided by the gear ratio. \[ \text{Rotation of Larger Gear} = \frac{270°}{1.971} \]

Step 5: Perform the division

\[ \text{Rotation of Larger Gear} \approx 137.0° \]

Final Answer

The larger gear rotates through approximately 137°.

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