A bicycle with 20-in.-diameter wheels has its gears set so that the chain has a 6-in. radius on the front sprocket and 4-in. radius on the rear sprocket. The cyclist pedals at 195 rpm. Round answers to 2 decimal places as needed. The linear speed of the bicycle in inches per minute = 24504.42 The speed of the bike in miles per hour = 23.20 xin/min 18378.32 mph o 17.4
A bicycle with 20-in.-diameter wheels has its gears set so that the chain has a 6-in. radius on the front sprocket and 4-in. radius on the rear sprocket. The cyclist pedals at 195 rpm. Round answers to 2 decimal places as needed. The linear speed of the bicycle in inches per minute = 24504.42 The speed of the bike in miles per hour = 23.20 xin/min 18378.32 mph o 17.4
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![### Bicycle Speed Calculation
#### Problem Statement:
A bicycle with 20-inch diameter wheels has its gears set so that the chain has a 6-inch radius on the front sprocket and a 4-inch radius on the rear sprocket. The cyclist pedals at 195 rpm. Calculate the linear speed of the bicycle in inches per minute and convert it to miles per hour.
**Note:** Round answers to 2 decimal places as needed.
---
#### Calculations:
1. **Linear Speed in inches per minute:**
Given:
- Front sprocket radius: 6 inches
- Rear sprocket radius: 4 inches
- Pedaling rate: 195 rpm
- Wheel diameter: 20 inches (hence, radius is 10 inches)
Using the relationship between the gear ratios and the wheel circumference to find the linear speed:
\[ \text{Linear speed} = \text{wheel radius} \times \text{pedaling rate} \times \left(\frac{\text{front sprocket radius}}{\text{rear sprocket radius}}\right) \times 2\pi \]
Substituting the values:
\[ \text{Linear speed} = 10 \times 195 \times \left(\frac{6}{4}\right) \times 2\pi \approx 24504.42 \, \text{in/min} \]
2. **Conversion to miles per hour:**
\[ \text{1 mile} = 63,360 \, \text{inches} \quad \text{and} \quad \text{1 hour} = 60 \, \text{minutes} \]
\[ \text{Speed in mph} = \left(\frac{\text{Speed in in/min} \times 60}{63360} \right) \]
Substituting the speed in inches per minute:
\[ \text{Speed in mph} = \left(\frac{24504.42 \times 60}{63360} \right) \approx 23.20 \, \text{mph} \]
Therefore:
- **Linear speed of the bicycle in inches per minute = 24,504.42 in/min**
- **Speed of the bike in miles per hour = 23.20 mph**
---
#### Verification:
For validation, alternatives using metric unit](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6aa8815b-de7d-43a5-aa48-91a8705759b5%2F7389637f-2e35-4bc2-ad30-afb03472af7c%2Frhs1zka_processed.png&w=3840&q=75)
Transcribed Image Text:### Bicycle Speed Calculation
#### Problem Statement:
A bicycle with 20-inch diameter wheels has its gears set so that the chain has a 6-inch radius on the front sprocket and a 4-inch radius on the rear sprocket. The cyclist pedals at 195 rpm. Calculate the linear speed of the bicycle in inches per minute and convert it to miles per hour.
**Note:** Round answers to 2 decimal places as needed.
---
#### Calculations:
1. **Linear Speed in inches per minute:**
Given:
- Front sprocket radius: 6 inches
- Rear sprocket radius: 4 inches
- Pedaling rate: 195 rpm
- Wheel diameter: 20 inches (hence, radius is 10 inches)
Using the relationship between the gear ratios and the wheel circumference to find the linear speed:
\[ \text{Linear speed} = \text{wheel radius} \times \text{pedaling rate} \times \left(\frac{\text{front sprocket radius}}{\text{rear sprocket radius}}\right) \times 2\pi \]
Substituting the values:
\[ \text{Linear speed} = 10 \times 195 \times \left(\frac{6}{4}\right) \times 2\pi \approx 24504.42 \, \text{in/min} \]
2. **Conversion to miles per hour:**
\[ \text{1 mile} = 63,360 \, \text{inches} \quad \text{and} \quad \text{1 hour} = 60 \, \text{minutes} \]
\[ \text{Speed in mph} = \left(\frac{\text{Speed in in/min} \times 60}{63360} \right) \]
Substituting the speed in inches per minute:
\[ \text{Speed in mph} = \left(\frac{24504.42 \times 60}{63360} \right) \approx 23.20 \, \text{mph} \]
Therefore:
- **Linear speed of the bicycle in inches per minute = 24,504.42 in/min**
- **Speed of the bike in miles per hour = 23.20 mph**
---
#### Verification:
For validation, alternatives using metric unit
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