Question

The length of the minute hand of a clock is 14cm. Find the area swept by the minute hand in 15 min.

Answer 1

Hint: Observe that the minute hand will form a sector of a circle. Find the angle of the sector and hence find the area of the sector using the formula area of a sector $=\dfrac{\theta }{360{}^\circ }\pi {{r}^{2}}$. Alternatively, find the area velocity of the minute hand and hence find the area swept in the given amount of time.

Complete step-by-step solution -

The minute hand of a clock sweeps $360{}^\circ $ in 60 minutes.
Hence the angle swept by the minute hand of the clock in 1 min $=\dfrac{360{}^\circ }{60}=6{}^\circ $
Hence the angle swept by the minute hand of the clock in 15 minutes $=6{}^\circ \times 15=90{}^\circ $
Hence the minute hand will sweep an area equal to the area of a sector of radius 14 cm and angle $90{}^\circ $.
We know that area of a sector of sectoral angle $\theta $ and radius r is given by $\dfrac{\theta }{360}\pi {{r}^{2}}$
Here $\theta =90{}^\circ $ and r =14cm.
Hence we have the area of the sector $=\dfrac{90}{360}\pi {{\left( 14 \right)}^{2}}=\dfrac{1}{4}\times \dfrac{22}{7}\times {{\left( 14 \right)}^{2}}=154$ square centimetres.
Hence the angle swept by the minute hand of the clock in 15 mins = 154 square centimetres.

Note: Alternative Solution:
We have the area swept by the minute hand of the clock in 60 minutes $=\pi {{r}^{2}}=\dfrac{22}{7}{{\left( 14 \right)}^{2}}=616$ square centimetres.
Hence the area swept by the minute hand clock in 1 min $=\dfrac{616}{60}$ square centimetres.
Hence the area swept by the minute hand of the clock in 15 minutes $=\dfrac{616}{60}\times 15=154$ square centimetres, which is the same as obtained above.