Question

At 3:40, the hour and minute hands of a clock are inclined at
(a) $\frac{2{\mathrm{\pi }}^{\mathrm{c}}}{3}$

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

(c) $\frac{13{\mathrm{\pi }}_{\mathrm{c}}}{18}$

(d) $\frac{13{\mathrm{\pi }}_{\mathrm{c}}}{4}$

Answer 1

(c) ${\left(\frac{13\mathrm{\pi }}{18}\right)}^c$
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,
$\mathrm{Angle}\mathrm{traced}\mathrm{by}\mathrm{the}\mathrm{hour}\mathrm{hand}\mathrm{in}3\mathrm{hours}40\mathrm{minutes},\mathrm{i}.\mathrm{e}.,\frac{11}{3}={\left(\frac{360}{12}\times \frac{11}{3}\right)}^{°}=110°$
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,
$\mathrm{Angle}\mathrm{traced}\mathrm{by}\mathrm{the}\mathrm{minute}\mathrm{hand}\mathrm{in}40\mathrm{minutes}={\left(\frac{360}{60}\times 40\right)}^{°}=240°$
∴ Required angle between two hands = $240°-110°=130°$
And,
Value of the angle (in radians) between the two hands of the clock = ${\left(130\times \frac{\mathrm{\pi }}{180}\right)}^c={\left(\frac{13\mathrm{\pi }}{18}\right)}^c=\frac{13{\mathrm{\pi }}^c}{18}$