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# help

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Find the angle between the hour hand and minute hand of a clock when it is 9:20.

Jun 16, 2020

#1
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At 9:40, the minute hand is at 4 and the hour hand is between 9 and 10.

There is $$30 ^\circ*5=150^\circ$$between the minute hand and 9 o'clock.

The hour hand is $$\frac{20}{60}*30^\circ=10^\circ$$ past 9 o'clock.

So our answer is $$150^\circ+10^\circ=160^\circ$$

Jun 16, 2020
#2
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Hour hand moves .5 degree per minute and starts at 270 degrees at nine o'clock

So hour hand is at.   270 + (20*.5)= 280o

Minute hand is at 120 degrees  at 20 after

280-120=160 between the hands....

Jun 16, 2020
#3
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Find the angle between the hour hand and minute hand of a clock when it is 9:20.

$$\begin{array}{rcll} \Delta\alpha^{\circ} &=& 330 \times t^h \pmod{360^{\circ}} \quad | \quad \mathbf{t^h=9+\dfrac{20}{60} = \dfrac{28}{3}\ h } \\ &=& 330 \times \dfrac{28}{3} \pmod{360^{\circ}} \\ &=& 3080^{\circ} \pmod{360^{\circ}} \\ &=& \mathbf{200^{\circ}} \qquad (\text{greater angle)} \\ &=& \mathbf{360^\circ-200^{\circ}=160^\circ} \qquad (\text{smaller angle)} \\ \end{array}$$ Jun 17, 2020