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?

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.
$x = \text{Distance}_\text{second hand} - \text{Distance}_\text{minute hand}$

The angular speed ($\omega$) is given by $\omega = \frac{2\pi}{\text{Period}}$.

• For the second hand: It completes one full revolution (2$\pi$ radians) in 60 seconds.

$$\omega_s = \frac{2\pi \text{ rad}}{60 \text{ s}} = \frac{\pi}{30} \text{ rad/s}$$

• For the minute hand: It completes one full revolution (2$\pi$ radians) in 60 minutes.

$$\omega_m = \frac{2\pi \text{ rad}}{60 \text{ min}} = \frac{2\pi \text{ rad}}{3600 \text{ s}} = \frac{\pi}{1800} \text{ rad/s}$$

The angle rotated ($\theta$) is given by $\theta = \omega T$.

• For the second hand:

$$\theta_s = \omega_s T = \left(\frac{\pi}{30} \text{ rad/s}\right) \times (1800 \text{ s}) = 60\pi \text{ rad}$$

• For the minute hand:

$$\theta_m = \omega_m T = \left(\frac{\pi}{1800} \text{ rad/s}\right) \times (1800 \text{ s}) = \pi \text{ rad}$$

The distance traveled (arc length, $L$) is given by $L = r\theta$.

• For the second hand:

$$L_s = r_s \theta_s = (75 \text{ cm}) \times (60\pi \text{ rad}) = 4500\pi \text{ cm}$$

• For the minute hand:

$$L_m = r_m \theta_m = (60 \text{ cm}) \times (\pi \text{ rad}) = 60\pi \text{ cm}$$

$$x = L_s - L_m = 4500\pi \text{ cm} - 60\pi \text{ cm}$$ $$x = (4500 - 60)\pi \text{ cm} = 4440\pi \text{ cm}$$

Substitute $\pi = 3.14$:

$$x = 4440 \times 3.14 \text{ cm} = 13941.6 \text{ cm}$$

Convert centimeters to meters ($1 \text{ m} = 100 \text{ cm}$):

$$x = \frac{13941.6}{100} \text{ m} = 139.416 \text{ m}$$

The approximate value of $x$ is 139.4 m.

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