The minute hand of a clock is 7 cm long. Find the area of the sector made by the minute hand between 7 a.m. and 7.05 a.m.

### Concept & Formula The minute hand of a clock sweeps a full circle ($360^\circ$) in 60 minutes. Therefore, the angle swept per minute is $360^\circ / 60 = 6^\circ$. The area swept is a sector of a circle with the length of the hand acting as the radius $r$. $$Area = \frac{\theta}{360^\circ} \times \pi r^2$$ ### Step-by-Step Solution * Given length (radius), $r = 7 \text{ cm}$. * Time elapsed between 7 a.m. and 7.05 a.m. is $5$ minutes. * Angle swept, $\theta = 5 \text{ mins} \times 6^\circ/\text{min} = 30^\circ$. * Area swept $= \frac{30}{360} \times \frac{22}{7} \times (7)^2$ * Area $= \frac{1}{12} \times \frac{22}{7} \times 49$ * Area $= \frac{1}{12} \times 22 \times 7 = \frac{154}{12}$ * Area $= \frac{77}{6} = 12.833... \text{ cm}^2$. ### Exam Strategy & Shortcut Instead of finding the angle, you can use the fraction of time directly. $5$ minutes is $\frac{5}{60} = \frac{1}{12}$ of an hour (a full circle). So the area is simply $\frac{1}{12}$ of the total circle area $\pi r^2$. $\frac{1}{12} \times \frac{22}{7} \times 49 = \frac{154}{12} = 12.83$. Since $12.83$ does not perfectly match $12.8$, "None of these" is the mathematically strict correct answer. ### Common Pitfall Students often select $12.8 \text{ cm}^2$ as an approximation. However, in standard aptitude tests featuring "None of these," exactness is paramount unless the question asks for an approximate value. $77/6$ is precisely $12.8\bar{3}$. ### Final Answer Therefore, the correct answer is **None of these**.