At 3:40, the Hour and Minute Hands of a Clock are Inclined at

Question:

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

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

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

(c) \(\frac{13\pi}{18}^c\)

(d) \(\frac{3\pi}{4}^c\)

Solution

At \(3:40\),

Minute hand position:

\[ 40 \times 6^\circ = 240^\circ \]

Hour hand position:

\[ 3 \times 30^\circ + 40 \times \frac{1}{2}^\circ \]

\[ = 90^\circ + 20^\circ \]

\[ = 110^\circ \]

Required angle:

\[ |240^\circ – 110^\circ| \]

\[ = 130^\circ \]

Converting into radians:

\[ 130^\circ = \frac{130\pi}{180} \]

\[ = \frac{13\pi}{18} \]

Hence, the correct option is:

(c) \( \frac{13\pi}{18} \)