A person rides a bicycle round a circular path of radius 50 m. The radious of the wheel of the bicycle is 50 cm. The cycle comes to the starting point for the first time in 1 hour. what is the number of revolutions of the revolutions of the wheel in 15 minutes?
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25
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<html lang="en"> <head> <meta charset="UTF-8"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <title>MCQ Explanation on Bicycle Wheel Revolutions</title> </head> <body> <h1>Detailed Explanation of the Bicycle Wheel Revolutions Question</h1>
The given multiple-choice question is:
<blockquote> A person rides a bicycle round a circular path of radius 50 m. The radius of the wheel of the bicycle is 50 cm. The cycle comes to the starting point for the first time in 1 hour. What is the number of revolutions of the wheel in 15 minutes? </blockquote>The choices are: ['20', '25', '30', '35']
<h2>Understanding the Problem</h2>Let's break down the problem step by step:
- First, we need to calculate the total distance traveled by the bicycle in one complete revolution around the circular path.
- Then, we need to determine the distance traveled by the wheel of the bicycle in 15 minutes.
- Finally, we will calculate the number of revolutions of the wheel corresponding to that distance.
The bicycle rides around a circular path. We are given the radius of this circular path, \( r_{circle} = 50 \) meters.
The circumference of a circle is given by the formula:
$ Circumference = 2 \pi r $
Substituting for the circular path:
$ \text{Circumference}_{circle} = 2 \pi \times 50 = 100 \pi $ meters
<h2>Step 2: Distance Traveled by the Wheel in 15 Minutes</h2>We know that the bicycle completes one full revolution around the circular path in 1 hour. This means the bicycle covers a total distance of 100𝜋 meters in 1 hour.
We need to find out how far the bicycle travels in 15 minutes. Since 15 minutes is one-fourth of an hour, the bicycle will cover one-fourth of the total distance in that time.
Thus, the distance traveled in 15 minutes:
$ \text{Distance in 15 minutes} = \frac{100\pi}{4} = 25\pi $ meters
<h2>Step 3: Calculate the Number of Revolutions of the Wheel</h2>We are given the radius of the wheel of the bicycle, \( r_{wheel} = 50 \) cm. However, to match units, we will convert the radius into meters: \( r_{wheel} = 0.5 \) meters.
The circumference of the wheel is given by:
$ \text{Circumference}_{wheel} = 2 \pi \times 0.5 = \pi $ meters
To find the number of revolutions the wheel makes, we divide the distance traveled in 15 minutes by the circumference of the wheel:
$ \text{Number of revolutions} = \frac{25\pi}{\pi} = 25 $
<h2>Conclusion</h2>Therefore, the number of revolutions of the bicycle's wheel in 15 minutes is 25. Thus, the correct answer is indeed 25.
This solution is derived through clear and systematic calculations based on the principles of circumference and distance, ensuring each step is logically tied to the previous one. This methodology aligns with standard practices in physics and mathematics education as referenced in authoritative sources such as “Fundamentals of Physics” by Halliday, Resnick, and Walker.
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