In a clock, second hand and minute hand are of 75 cm and 60 cm respectively. After 30 minutes, ratio of distance travelled by the tip of second hand to that of minute hand is x. Find x.
To find the ratio \( x \) of the distance travelled by the tips of the second hand and the minute hand after 30 minutes, we proceed as follows:
1. **Calculate the distance travelled by the tip of the second hand:** The second hand completes 1 full revolution per minute. Therefore, in 30 minutes, it completes \( 30 \times 1 = 30 \) revolutions. Each revolution is equivalent to the circumference of the circle that the tip of the second hand traces. This is given by the formula \( 2\pi r \), where \( r = 75 \text{ cm} \). Thus, distance travelled by the second hand is: \( 30 \times 2\pi \times 75 = 4500\pi \).
2. **Calculate the distance travelled by the tip of the minute hand:** The minute hand completes 1 full revolution per hour (60 minutes). Therefore, in 30 minutes, it completes \( \frac{30}{60} = 0.5 \) revolutions. The circumference of the circle traced by the tip of the minute hand is \( 2\pi \times 60 \). Thus, the distance travelled by the minute hand is: \( 0.5 \times 2\pi \times 60 = 60\pi \).
3. **Compute the ratio \( x \):** Ratio \( x = \frac{\text{Distance travelled by the second hand}}{\text{Distance travelled by the minute hand}} = \frac{4500\pi}{60\pi} = \frac{4500}{60} = 75 \).
4. **Verify the computed value against the given range:** Since the expected range is 75,75, the computed value of \( x = 75 \) falls within this range.
Therefore, the ratio of the distance travelled by the tip of the second hand to that of the minute hand after 30 minutes is \( 75 \).
