Question

The length of the minute hand of a clock is 14 cm. Find the area swept by the minute hand in 5 minutes.

Answer 1

The minute hand of a clock completes a full rotation of \(360°\) in 60 minutes (1 hour).

In 5 minutes, the minute hand rotates by an angle of \(\frac{360°}{60} \times 5 = 30°\).

The area swept by the minute hand in 5 minutes is the area of a sector with a central angle of \(30°\) in a circle of radius \(14 \,cm\).

The formula for the area of a sector with angle \(θ\) is \(\frac{θ}{360°} \times π r^2\).

For a \(30°\) sector with a radius of \(14 \,cm\), the area is \(\frac{30°}{360°} \times \frac{22}{7} \times 14 \times 14\).

This simplifies to \(\frac{1}{12} \times \frac{22}{7} \times 196 = \frac{22}{12} \times 28 = \frac{11}{6} \times 28 = \frac{11}{3} \times 14 = \frac{154}{3} \,cm^2\).

Therefore, the area swept by the minute hand in \(5\) minutes is \(\frac{154}{3}\, cm^2\).