Question

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

• A. $\frac{2{\pi }^c}{3}$
• B. $\frac{7{\pi }^c}{12}$
• C. $\frac{13{\pi }^c}{18}$
• D. $\frac{3{\pi }^c}{4}$

Answer 1

The correct option is C

$\frac{13{\pi }^c}{18}$

We know that the hour hand of a clock completes one rotation in 12 hours.

$\therefore$ Angle traced by the hour hand in 12 hours $={360}^{\circ }$

Now,

Angle traced by the hour hand in 3 hours 40 minutes, i.e.,

$\frac{11}{3}={(\frac{360}{12}\times \frac{11}{3})}^{\circ }$

$={110}^{\circ }$

We also know that the minute hand of a clock completes one rotation in 60 minutes.

$\therefore$ Angle traced by the minute hand in 60 minutes = ${360}^{\circ }$

Now,

Angle traced by the minute hand in 40 minutes $=(\frac{360}{60}\times {40}^{\circ })={240}^{\circ }$

$\therefore$ 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

$={(130\times \frac{\pi }{180})}^c={\left(\frac{13\pi }{18}\right)}^c=\frac{13{\pi }^c}{18}$