A clock has second and minute hands of lengths 7.5 cm and 6.0 cm respectively. What is the ratio of the distance covered by the tip of the minute hand in 30 minutes to that covered by the tip of the second hand in 30 minutes?

Step 1: Identify the given information and the goal. Given: • Length of second hand, $r_s = 75 \text{ cm}$ • Length of minute hand, $r_m = 60 \text{ cm}$ • Time duration, $T = 30 \text{ minutes}$ • Value of $\pi = 3.14$

Goal: Find the difference $x$ in the distance traveled by the tip of the second hand and the tip of the minute hand in meters. $x = Distance_second hand - Distance_\text{minute hand}$

Step 2: Convert the time duration to seconds. $T = 30 minutes \times 60 seconds/minute = 1800 seconds$

Step 3: Calculate the angular speed of each hand. The angular speed ($\omega$) is given by $\omega = \frac{2\pi}{Period}$. • For the second hand: It completes one full revolution (2$\pi$ radians) in 60 seconds. $\omega_s = \frac{2\pi rad}{60 s} = \frac{\pi}{30} rad/s$ • For the minute hand: It completes one full revolution (2$\pi$ radians) in 60 minutes. $\omega_m = \frac{2\pi rad}{60 min} = \frac{2\pi rad}{3600 s} = \frac{\pi}{1800} rad/s$

Step 4: Calculate the angle rotated by each hand in $T = 1800$ seconds. The angle rotated ($\theta$) is given by $\theta = \omega T$. • For the second hand: $\theta_s = \omega_s T = \left(\frac{\pi}{30} rad/s\right) \times (1800 s) = 60\pi rad$ • For the minute hand: $\theta_m = \omega_m T = \left(\frac{\pi}{1800} rad/s\right) \times (1800 s) = \pi rad$

Step 5: Calculate the distance traveled by the tip of each hand. The distance traveled (arc length, $L$) is given by $L = r\theta$. • For the second hand: $L_s = r_s \theta_s = (75 cm) \times (60\pi rad) = 4500\pi cm$ • For the minute hand: $L_m = r_m \theta_m = (60 cm) \times (\pi rad) = 60\pi cm$

Step 6: Calculate the difference $x$ and convert it to meters. $x = L_s - L_m = 4500\pi cm - 60\pi cm$ $x = (4500 - 60)\pi cm = 4440\pi cm$ Substitute $\pi = 3.14$: $x = 4440 \times 3.14 cm = 13941.6 cm$ Convert centimeters to meters ($1 m = 100 \text{ cm}$): $x = \frac{13941.6}{100} m = 139.416 m$

The approximate value of $x$ is 139.4 m.

The final answer is $\boxed{139.4 m}$.