The length of the minute hand of a clock is 5 cm. Find the area swept by it from 6:05 a.m. to 6:40 a.m.
\(\dfrac{175}{12}\,\pi\,\text{cm}^2 \;\approx\; 45.8\,\text{cm}^2\)
Step 1: The length of the minute hand is given as 5 cm. This is the radius of the circular path made by the tip of the hand. Radius \(r = 5\,\text{cm} = 0.05\,\text{m}\) (converted to SI unit, metres).
Step 2: The time interval is from 6:05 a.m. to 6:40 a.m. That is a total of \(40 - 5 = 35\) minutes.
Step 3: The minute hand completes one full circle (360°) in 60 minutes. So, in 35 minutes it covers: \(\dfrac{35}{60} \times 360^{\circ} = 210^{\circ}\).
Step 4: The area swept by the hand is the area of a sector of a circle. Formula for sector area: \[ A = \dfrac{\theta}{360^{\circ}} \times \pi r^2 \]
Step 5: Substituting values: \[ A = \dfrac{210}{360} \times \pi (0.05)^2 \, \text{m}^2 \]
Step 6: Simplify: \[ A = \dfrac{7}{12} \times \pi (0.0025) \, \text{m}^2 = \dfrac{0.0175}{12} \pi \, \text{m}^2 \]
Converting back to cm² (since 1 m² = 10,000 cm²): \[ A = \dfrac{175}{12} \pi \, \text{cm}^2 \approx 45.8 \, \text{cm}^2 \]
Final Answer: The area swept by the minute hand is about \(45.8 \, \text{cm}^2\).