Let's solve this problem step by step.

The question is about a clock's minute hand that has moved for 30 minutes, and we need to find the distance traveled by the tip of the minute hand in centimeters.

Given Information

1. The length of the minute hand is $14 \, \text{cm}$.
2. The minute hand moves for 30 minutes.

Solution

1. Determine the Angle Covered: In 60 minutes, the minute hand completes a full circle, which is $360^\circ$.

   • Since 30 minutes is half of 60 minutes, the minute hand will cover half of $360^\circ$, which is $180^\circ$.

2. Calculate the Arc Length:

   • The arc length $L$ for an angle $\theta$ in a circle of radius $r$ is given by: $L = \frac{\theta}{360^\circ} \times 2 \pi r$
   • Here, $\theta = 180^\circ$ and $r = 14 \, \text{cm}$.

3. Substitute the Values: $L = \frac{180}{360} \times 2 \times \pi \times 14$ Simplifying this: $L = \frac{1}{2} \times 2 \pi \times 14$ $L = \pi \times 14$ $L = 14 \pi \, \text{cm}$

4. Approximate the Answer (using $\pi \approx 3.14$): $L \approx 14 \times 3.14$ $L \approx 43.96 \, \text{cm}$

Final Answer

The distance traveled by the tip of the minute hand is approximately 43.96 cm.

Would you like further details on any part of this solution, or do you have any additional questions?

Here are some related questions to expand your understanding:

1. How would the answer change if the minute hand length was different?
2. What would be the distance covered if the minute hand moved for 15 minutes?
3. How can we generalize this formula for any movement of the minute hand?
4. What if we were calculating the path of the hour hand instead?
5. How does the concept of angular displacement apply here?

Tip: Remember that a minute hand covers $6^\circ$ per minute because $360^\circ$ is divided among 60 minutes.