The figure shows the chain drive of a bicycle. How far will the bicycle move if the pedals are rotated through 180°? Assume the radius of the bicycle wheel is 12.3 inches. 151 in 4.28 in
The figure shows the chain drive of a bicycle. How far will the bicycle move if the pedals are rotated through 180°? Assume the radius of the bicycle wheel is 12.3 inches. 151 in 4.28 in
Trigonometry (MindTap Course List)
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Author:Ron Larson
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![**Bicycle Chain Drive Mechanics**
*The figure shows the chain drive of a bicycle. How far will the bicycle move if the pedals are rotated through 180°? Assume the radius of the bicycle wheel is 12.3 inches.*
**Explanation:**
The image depicts the mechanics of a bicycle's chain drive system. Here is a detailed description of what is displayed:
- There is a large sprocket, which has a marked radius of **4.28 inches**.
- A smaller sprocket connected by a chain to the large sprocket with a radius of **1.51 inches**.
- The circle represents the wheel of the bicycle with an assumed radius of **12.3 inches**.
**Steps to Calculate the Distance Moved by the Bicycle:**
1. **Rotation of Pedals and Sprocket Ratio:**
- The pedals are rotated through 180°.
- 180° of pedal rotation means the large sprocket completes 0.5 rotations because 180° equals half a full circle.
2. **Distance the Chain Travels:**
- The circumference of the larger sprocket, given that Circumference \( C = 2 * \pi * r \):
\[ C_{large\ sprocket} = 2 * \pi * 4.28 \approx 26.89\ inches \]
- For 0.5 rotations:
\[ 0.5 * 26.89 \approx 13.45\ inches \]
3. **Distance Moved by Rear Wheel:**
- The rear sprocket with a radius of 1.51 inches also rotates as the large sprocket turns.
- The ratio of the radii of the large sprocket to the small sprocket determines how many times the rear wheel rotates for each turn of the pedals:
\[ \text{Rotation Ratio} = \frac{4.28}{1.51} \approx 2.83 \]
- So, each rotation of the large sprocket moves the wheel approximately 2.83 times more.
- For 0.5 rotations of the larger sprocket, the smaller sprocket rotates:
\[ 0.5 * 2.83 \approx 1.415 \]
4. **Wheel Circumference and Distance Travelled by Bicycle:**
- The bicycle wheel's circumference:
\[ C_{wheel} =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd9470fd7-7966-4363-9efa-190bda48f253%2F58f55205-a870-40df-8122-2165e2017440%2Fgws71r_processed.png&w=3840&q=75)
Transcribed Image Text:**Bicycle Chain Drive Mechanics**
*The figure shows the chain drive of a bicycle. How far will the bicycle move if the pedals are rotated through 180°? Assume the radius of the bicycle wheel is 12.3 inches.*
**Explanation:**
The image depicts the mechanics of a bicycle's chain drive system. Here is a detailed description of what is displayed:
- There is a large sprocket, which has a marked radius of **4.28 inches**.
- A smaller sprocket connected by a chain to the large sprocket with a radius of **1.51 inches**.
- The circle represents the wheel of the bicycle with an assumed radius of **12.3 inches**.
**Steps to Calculate the Distance Moved by the Bicycle:**
1. **Rotation of Pedals and Sprocket Ratio:**
- The pedals are rotated through 180°.
- 180° of pedal rotation means the large sprocket completes 0.5 rotations because 180° equals half a full circle.
2. **Distance the Chain Travels:**
- The circumference of the larger sprocket, given that Circumference \( C = 2 * \pi * r \):
\[ C_{large\ sprocket} = 2 * \pi * 4.28 \approx 26.89\ inches \]
- For 0.5 rotations:
\[ 0.5 * 26.89 \approx 13.45\ inches \]
3. **Distance Moved by Rear Wheel:**
- The rear sprocket with a radius of 1.51 inches also rotates as the large sprocket turns.
- The ratio of the radii of the large sprocket to the small sprocket determines how many times the rear wheel rotates for each turn of the pedals:
\[ \text{Rotation Ratio} = \frac{4.28}{1.51} \approx 2.83 \]
- So, each rotation of the large sprocket moves the wheel approximately 2.83 times more.
- For 0.5 rotations of the larger sprocket, the smaller sprocket rotates:
\[ 0.5 * 2.83 \approx 1.415 \]
4. **Wheel Circumference and Distance Travelled by Bicycle:**
- The bicycle wheel's circumference:
\[ C_{wheel} =
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