1. **Problem statement:** We have a clock with minute and second hands each 1 cm long. We want to find the angular displacement (Winkelgröße) and arc length (Bogenlänge) traveled by the tips of these hands for various elapsed times. 2. **Formulas and rules:** - The full circle is 360° or $2\pi$ radians. - The minute hand completes one full rotation (360°) in 60 minutes. - The second hand completes one full rotation (360°) in 60 seconds (1 minute). - Angular displacement $\theta$ in degrees for the minute hand after $t$ minutes is $\theta = 6t$ degrees because $360°/60 = 6°$ per minute. - Angular displacement $\theta$ in degrees for the second hand after $t$ minutes is $\theta = 360t$ degrees because it completes 1 rotation per minute. - Arc length $s$ is given by $s = r \times \theta_{rad}$ where $\theta_{rad}$ is the angle in radians. - To convert degrees to radians: $\theta_{rad} = \theta \times \frac{\pi}{180}$. 3. **Calculations for each time:** **a) 12 minutes** - Minute hand angle: $\theta_m = 6 \times 12 = 72°$ - Second hand angle: $\theta_s = 360 \times 12 = 4320°$ - Convert to radians: $$\theta_{m,rad} = 72 \times \frac{\pi}{180} = \frac{2\pi}{5}$$ $$\theta_{s,rad} = 4320 \times \frac{\pi}{180} = 24\pi$$ - Arc lengths (radius $r=1$ cm): $$s_m = 1 \times \frac{2\pi}{5} = \frac{2\pi}{5} \approx 1.257$$ cm $$s_s = 1 \times 24\pi = 24\pi \approx 75.398$$ cm **b) 59 minutes** - Minute hand angle: $\theta_m = 6 \times 59 = 354°$ - Second hand angle: $\theta_s = 360 \times 59 = 21240°$ - Convert to radians: $$\theta_{m,rad} = 354 \times \frac{\pi}{180} = \frac{59\pi}{30}$$ $$\theta_{s,rad} = 21240 \times \frac{\pi}{180} = 118\pi$$ - Arc lengths: $$s_m = 1 \times \frac{59\pi}{30} = \frac{59\pi}{30} \approx 6.180$$ cm $$s_s = 1 \times 118\pi = 118\pi \approx 370.884$$ cm **c) 24 minutes** - Minute hand angle: $\theta_m = 6 \times 24 = 144°$ - Second hand angle: $\theta_s = 360 \times 24 = 8640°$ - Convert to radians: $$\theta_{m,rad} = 144 \times \frac{\pi}{180} = \frac{4\pi}{5}$$ $$\theta_{s,rad} = 8640 \times \frac{\pi}{180} = 48\pi$$ - Arc lengths: $$s_m = 1 \times \frac{4\pi}{5} = \frac{4\pi}{5} \approx 2.513$$ cm $$s_s = 1 \times 48\pi = 48\pi \approx 150.796$$ cm **d) 130 minutes** - Minute hand angle: $\theta_m = 6 \times 130 = 780°$ - Second hand angle: $\theta_s = 360 \times 130 = 46800°$ - Convert to radians: $$\theta_{m,rad} = 780 \times \frac{\pi}{180} = \frac{13\pi}{3}$$ $$\theta_{s,rad} = 46800 \times \frac{\pi}{180} = 260\pi$$ - Arc lengths: $$s_m = 1 \times \frac{13\pi}{3} = \frac{13\pi}{3} \approx 13.613$$ cm $$s_s = 1 \times 260\pi = 260\pi \approx 816.814$$ cm **e) 4.5 hours = 270 minutes** - Minute hand angle: $\theta_m = 6 \times 270 = 1620°$ - Second hand angle: $\theta_s = 360 \times 270 = 97200°$ - Convert to radians: $$\theta_{m,rad} = 1620 \times \frac{\pi}{180} = 9\pi$$ $$\theta_{s,rad} = 97200 \times \frac{\pi}{180} = 540\pi$$ - Arc lengths: $$s_m = 1 \times 9\pi = 9\pi \approx 28.274$$ cm $$s_s = 1 \times 540\pi = 540\pi \approx 1696.460$$ cm **f) 8.25 hours = 495 minutes** - Minute hand angle: $\theta_m = 6 \times 495 = 2970°$ - Second hand angle: $\theta_s = 360 \times 495 = 178200°$ - Convert to radians: $$\theta_{m,rad} = 2970 \times \frac{\pi}{180} = \frac{33\pi}{2}$$ $$\theta_{s,rad} = 178200 \times \frac{\pi}{180} = 990\pi$$ - Arc lengths: $$s_m = 1 \times \frac{33\pi}{2} = \frac{33\pi}{2} \approx 51.836$$ cm $$s_s = 1 \times 990\pi = 990\pi \approx 3109.008$$ cm **Summary:** - The minute hand angle is $\theta_m = 6t$ degrees for $t$ in minutes. - The second hand angle is $\theta_s = 360t$ degrees for $t$ in minutes. - Arc length is $s = r \times \theta_{rad}$ with $r=1$ cm. This completes the calculations for all given times.