The length of the minutes hand of a wall clock is 6 cm. Find the distance covered by the tip of the minutes hand in 25 minutes

Options

A. `(270)/(1)cm`
B. `110 cm `
C. `(88)/(7)cm `
D. ` 110/7 cm `

Explanation

To find the distance covered by the tip of the minute hand in 25 minutes, we can follow these steps:

### Step 1: Determine the angle covered by the minute hand in 25 minutes.
The minute hand completes a full rotation (360 degrees) in 60 minutes. Therefore, in 25 minutes, the angle covered can be calculated as follows:

\[ \text{Angle covered} = \left( \frac{25}{60} \right) \times 360 \]

### Step 2: Calculate the angle.
Now, let's calculate the angle:

\[ \text{Angle covered} = \left( \frac{25}{60} \right) \times 360 = \frac{25 \times 360}{60} = 150 \text{ degrees} \]

### Step 3: Find the distance covered by the tip of the minute hand.
The distance covered by the tip of the minute hand can be calculated using the formula for the arc length, which is given by:

\[ \text{Arc Length} = \frac{\theta}{360} \times 2\pi r \]

Where:
- \(\theta\) is the angle in degrees (150 degrees),
- \(r\) is the radius (length of the minute hand, which is 6 cm).

### Step 4: Substitute the values into the formula.
Now, substituting the values into the arc length formula:

\[ \text{Arc Length} = \frac{150}{360} \times 2\pi \times 6 \]

### Step 5: Simplify the expression.
Calculating this step-by-step:

1. Calculate \(\frac{150}{360}\):

\[ \frac{150}{360} = \frac{5}{12} \]

2. Calculate \(2\pi \times 6\):

\[ 2\pi \times 6 = 12\pi \]

3. Now substitute back into the arc length formula:

\[ \text{Arc Length} = \frac{5}{12} \times 12\pi = 5\pi \]

### Step 6: Calculate the numerical value.
Using \(\pi \approx 3.14\):

\[ \text{Arc Length} \approx 5 \times 3.14 = 15.7 \text{ cm} \]

### Final Answer:
The distance covered by the tip of the minute hand in 25 minutes is approximately **15.7 cm**.

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