Question

How much distance (in cm) will the bicycle shown below travel, if the pedal makes 1.5 revolutions? (Assume \(\pi = 22/7\)).
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Hint:

In bicycle motion problems, identify the key components: the pedal, the front sprocket, the rear sprocket, and the rear wheel. The front wheel's size is usually irrelevant unless the question is about stability or other dynamics. The distance is always tied to the rear wheel's rotation.

Answer 1

Step 1: Understanding the Concept:
The distance a bicycle travels is determined by the circumference of its driving wheel (the rear wheel) and the number of times that wheel rotates. The rotation of the rear wheel is linked to the rotation of the pedals through the gear system (sprockets and chain). Step 2: Key Formula or Approach:
1. Gear Ratio: The ratio of rotation between the rear wheel and the pedal is the gear ratio. \[ \text{Gear Ratio} = \frac{\text{Diameter of front sprocket}}{\text{Diameter of rear sprocket}} \] 2. Rear Wheel Revolutions: \[ \text{Wheel Revolutions} = \text{Pedal Revolutions} \times \text{Gear Ratio} \] 3. Distance Traveled: \[ \text{Distance} = \text{Wheel Revolutions} \times \text{Circumference of Rear Wheel} \] 4. Circumference: \[ \text{Circumference} = \pi \times \text{Diameter of Rear Wheel} \] Step 3: Detailed Explanation:
From the diagram, we have the following information:
- Diameter of front sprocket (pedal gear) = 10 cm.
- Diameter of rear sprocket (wheel gear) = 5 cm.
- Diameter of rear wheel = 35 cm.
- Number of pedal revolutions = 1.5.
Calculation 1: Gear Ratio
\[ \text{Gear Ratio} = \frac{10 \text{ cm}}{5 \text{ cm}} = 2 \] This means for every one revolution of the pedal, the rear wheel makes two revolutions. Calculation 2: Rear Wheel Revolutions
\[ \text{Wheel Revolutions} = 1.5 \times 2 = 3 \] The rear wheel will make 3 complete revolutions. Calculation 3: Circumference of Rear Wheel
\[ \text{Circumference} = \pi \times d = \frac{22}{7} \times 35 \text{ cm} = 22 \times 5 \text{ cm} = 110 \text{ cm} \] Calculation 4: Total Distance Traveled
\[ \text{Distance} = \text{Wheel Revolutions} \times \text{Circumference} = 3 \times 110 \text{ cm} = 330 \text{ cm} \] Step 4: Final Answer:
The total distance the bicycle will travel is 330 cm. This value falls within the specified correct answer range of 329.7 - 330.3.