Question

At 3:40, the hour and minute hands of a clock are inclined at

Options

• \[\frac{2 \pi^c}{3}\]

   

• \[\frac{7 \pi^c}{12}\]

   

• \[\frac{13 \pi_c}{18}\]

   

• \[\frac{13 \pi_c}{4}\]

   

Answer 1

\[\frac{13 \pi_c}{18}\]
We know that the hour hand of a clock completes one rotation in 12 hours.
∴ Angle traced by the hour hand in 12 hours = 360°
Now,
Angle traced by the hour hand in 3 hours 40 minutes, i . e . , \[\frac{11}{3} = \left( \frac{360}{12} \times \frac{11}{3} \right)^\circ= 110^\circ\]
We also know that the minute hand of a clock completes one rotation in 60 minutes.
∴ Angle traced by the minute hand in 60 minutes = 360°
Now,
Angle traced by the minute hand in 40 minutes = \[\left( \frac{360}{60} \times 40 \right)^\circ = 240^\circ\]
∴ Required angle between two hands = \[240^\circ - 110^\circ = 130^\circ\]
And,
Value of the angle (in radians) between the two hands of the clock = \[\left( 130 \times \frac{\pi}{180} \right)^c = \left( \frac{13\pi}{18} \right)^c = \frac{13 \pi^c}{18}\]