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Question Number 210875 by peter frank last updated on 20/Aug/24

	Answered by mahdipoor last updated on 20/Aug/24

	$g e t t i m e i s h : m & v e r t i c a l l i n e i s r e f r e n c e$ $a n g l e h u u r_h a n d : \frac{360}{12} h + \frac{360}{12 \times 60} m$ $a n g l e m i n u t e_h a n d : \frac{360}{60} m$ $\Rightarrow ∠ =∣ h - m ∣=∣ 30 h - 5.5 m ∣ o r 360 - ∣ 30 h - 5.5 m ∣$ $f o r e x a m p l e : 5 : 45$ $\Rightarrow∣ 30 \times 5 - 5.5 \times 45 ∣= 97.5 d e g = 1.702 r a d$
	Answered by efronzo1 last updated on 20/Aug/24

	$θ = \frac{∣ 60.5 - 11.45 ∣^{o}}{2} = 97 . 5^{o}$
	Answered by mr W last updated on 20/Aug/24

	$a s s u m e t h a t t h e h a n d s o f t h e c l o c k$ $m o v e \underset{―}{c o n t i n u o u s l y} .$ $a t t i m e h : 00$ $θ_{h} = \frac{h}{12} \times 360 = 30 h (°)$ $θ_{m} = 0 °$ $a t t i m e h : m$ $θ_{h} = 30 (h + \frac{m}{60}) = 30 h + \frac{m}{2} (°)$ $θ_{m} = \frac{m}{60} \times 360 = 6 m (°)$ $Δ θ = θ_{m} - θ_{h} = \frac{11 m}{2} - 30 h (°)$ $e x a m p l e a t t i m e 5 : 45$ $Δ θ = \frac{11 \times 45}{2} - 30 \times 5 = 97.5 (°)$ $e x a m p l e a t t i m e 10 : 12$ $Δ θ = \frac{11 \times 12}{2} - 30 \times 10 = - 234 (°) = 126 °$
	Commented by peter frank last updated on 22/Aug/24

	$thank you$
	Answered by mr W last updated on 20/Aug/24

	Commented by mr W last updated on 21/Aug/24

	$n o w {l e t}^{'} s l o o k a t a r e a l c l o c k .$ $o n a m e c h a n i c a l l y d r i v e n c l o c k t h e$ $h a n d s {d o n}^{'} t m o v e c o n t i n u o u s l y .$ $t h e y c a n o n l y h a v e 60 p o s s i b l e$ $p o s i t i o n s a r o u n d t h e c i r c l e . f o r$ $e x a m p l e f r o m 3 : 00 t o 3 : 12 t h e h o u r$ $h a n d r e m a i n s i n t h e s a m e p o s i t i o n$ $a s a t 3 : 00 . t h e p i c t u r e a b o v e s h o w s$ $e . g . t h e t i m e 3 : 07 : 03 .$ $i n f o l l o w i n g p i c t u r e t h e r e d h o u r$ $h a n d a n d r e d m i n u t e h a n d s h o w$ $t h e t i m e 5 : 45 . t h e a n g l e b e t w e e n$ $t h e m i s 17 \times 6 = 102 ° (\neq 97.5 °) .$
	Commented by mr W last updated on 20/Aug/24

	Commented by mr W last updated on 20/Aug/24

	$f o l l o w i n g p i c t u r e s h o w s t h e t i m e$ $10 : 12 . t h e a n g l e b e t w e e n t h e m i n u t e$ $h a n d a n d h o u r h a n d i s$ $21 \times 6 = 126 °$
	Commented by mr W last updated on 20/Aug/24

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