The large hand of a clock is 42 cm long. How many centimetres does its extremity move in 20 minutes.

Answer

To solve the problem of how many centimeters the extremity of the large hand of a clock moves in 20 minutes, we can follow these steps:

### Step 1: Understand the Movement of the Clock Hand
The large hand of the clock (minute hand) makes a complete revolution (360 degrees) in 60 minutes. Therefore, in 20 minutes, it will cover a fraction of the total angle.

### Step 2: Calculate the Angle Covered in 20 Minutes
The angle covered by the minute hand in 20 minutes can be calculated as follows:
\[ \text{Angle covered} = \left(\frac{20 \text{ minutes}}{60 \text{ minutes}}\right) \times 360^\circ = \frac{1}{3} \times 360^\circ = 120^\circ \]

### Step 3: Convert the Angle to Radians
To find the arc length (distance moved), we can convert the angle from degrees to radians. The conversion factor is:
\[ 1 \text{ radian} = \frac{180^\circ}{\pi} \]
Thus,
\[ 120^\circ = \frac{120 \times \pi}{180} = \frac{2\pi}{3} \text{ radians} \]

### Step 4: Use the Arc Length Formula
The formula for arc length \( s \) is given by:
\[ s = r \theta \]
where \( r \) is the radius (length of the minute hand) and \( \theta \) is the angle in radians. Here, \( r = 42 \text{ cm} \) and \( \theta = \frac{2\pi}{3} \text{ radians} \).

Substituting the values:
\[ s = 42 \times \frac{2\pi}{3} \]

### Step 5: Calculate the Arc Length
Now, we can calculate the arc length:
\[ s = 42 \times \frac{2 \times 3.14}{3} \approx 42 \times 2.094 \approx 88 \text{ cm} \]

Thus, the extremity of the large hand of the clock moves approximately **88 cm** in 20 minutes.

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