The distance covered by the tip of a minute hand in 35 minutes is 33 cm. What is the length of the minute hand ?

Options

A. 6 cm
B. 9 cm
C. 10 cm
D. 12 cm

Explanation

To find the length of the minute hand based on the distance it covers in a given time, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: The minute hand covers a distance of 33 cm in 35 minutes. We need to find the length of the minute hand (r).

2. **Determine the Angle Covered**: The minute hand moves 360 degrees in 60 minutes. Therefore, in 1 minute, it moves:
\[ \text{Degrees per minute} = \frac{360 \text{ degrees}}{60 \text{ minutes}} = 6 \text{ degrees} \]
In 35 minutes, the angle covered (θ) is:
\[ \theta = 35 \text{ minutes} \times 6 \text{ degrees/minute} = 210 \text{ degrees} \]

3. **Convert Degrees to Radians**: To use the formula for arc length, we need to convert degrees to radians. The conversion factor is:
\[ 1 \text{ degree} = \frac{\pi}{180} \text{ radians} \]
Thus, converting 210 degrees to radians:
\[ \theta = 210 \times \frac{\pi}{180} = \frac{7\pi}{6} \text{ radians} \]

4. **Use the Arc Length Formula**: The distance (s) covered by the tip of the minute hand can be expressed as:
\[ s = r \cdot \theta \]
Where \(s\) is the distance covered (33 cm) and \(r\) is the length of the minute hand. Rearranging the formula to solve for \(r\):
\[ r = \frac{s}{\theta} \]

5. **Substitute the Values**: Now substitute \(s = 33 \text{ cm}\) and \(\theta = \frac{7\pi}{6}\):
\[ r = \frac{33}{\frac{7\pi}{6}} = 33 \times \frac{6}{7\pi} \]

6. **Calculate the Length of the Minute Hand**: Simplifying the expression:
\[ r = \frac{198}{7\pi} \]
Using \(\pi \approx 3.14\):
\[ r \approx \frac{198}{21.99} \approx 9 \text{ cm} \]

### Final Answer:
The length of the minute hand is approximately **9 cm**.