**Question:**

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

(a) $\frac{2 \pi^{c}}{3}$

(b) $\frac{7 \pi^{\mathrm{c}}}{12}$

(C) $\frac{13 \pi_{\mathrm{c}}}{18}$

(d) $\frac{13 \pi_{c}}{4}$

**Solution:**

(C) $\frac{13 \pi_{\mathrm{c}}}{18}$

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 =

And,

Value of the angle (in radians) between the two hands of the clock $=\left(130 \times \frac{\pi}{180}\right)^{c}=\left(\frac{13 \pi}{18}\right)^{c}=\frac{13 \pi^{c}}{18}$

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