Question

Find the time between 3 and 4 O’ clock when
the angle between the hands of a watch is one third of a right angle.
(1) min past 3
(2) min past 3
(3) min past 3
(4) min past 3

Answer 1

Speed of hour hand as it moves \frac{360}{12}

12

360

=30^{\circ} \text { in one hour }=30

∘

in one hour

That is in 60 minutes hour hand moves 30^{\circ}30

∘

So in 1 minute hour hand moves \frac{30}{60}

60

30

= \frac{1}{2}

2

1

degrees

So in x minutes hour will move \frac{1}{2} \mathrm{x}=\frac{\mathrm{x}}{2} \text { degrees }

2

1

x=

2

x

degrees

At 3 o clock, hour hand is 45 degree ahead of minute hand.

So after 3 o clock and x minutes, position of hour hand = \frac{x}{2}

2

x

+ 45 degree

In 60 minutes, minute hand moves 360^{\circ}360

∘

In 1 minutes, minute hand moves \frac{360}{60}=6^{\circ}

60

360

=6

∘

So in x minutes , minutes hand moves =6 x^{0}=6x

0

We want that value of x when:

one third of right angle is 30^{\circ}30

∘

\left(\frac{x}{2}+45\right)-6 x=30(

2

x

+45)−6x=30

x + 90 -12x = 60

-11x = 60 – 90 = -30

x = \frac{30}{11} = 2.7272x=

11

30

=2.7272

So at 3 o clock and 2.7272 minutes, angle between the hands of a watch is one-third of a right angle.