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Question Number 25895 by Mr eaay last updated on 16/Dec/17
The hands of a clock have length 3cm   and 5cm.Calculate the distance between  the tips of the hand at 4 “0” clock.
$${The}\:{hands}\:{of}\:{a}\:{clock}\:{have}\:{length}\:\mathrm{3}{cm}\: \\ $$$${and}\:\mathrm{5}{cm}.{Calculate}\:{the}\:{distance}\:{between} \\ $$$${the}\:{tips}\:{of}\:{the}\:{hand}\:{at}\:\mathrm{4}\:“\mathrm{0}''\:{clock}. \\ $$
Answered by mrW1 last updated on 16/Dec/17
angle between the hands:  at h hours and m minutes  θ=(((h+(m/(60)))/(12))−(m/(60)))×360°=((h/(12))−((11m)/(720)))×360°    at 4:00  ⇒θ=(4/(12))×360=120°  distance:  d=(√(l_H ^2 +l_M ^2 −2l_H l_M cos θ))  ⇒d=(√(3^2 +5^2 −2×3×5×(−0.5)))=7 cm    at 4:40  θ=((4/(12))−((11×40)/(720)))×360°=−100°  ⇒d=(√(3^2 +5^2 −2×3×5×cos (−100°)))=6.26 cm
$${angle}\:{between}\:{the}\:{hands}: \\ $$$${at}\:\boldsymbol{{h}}\:{hours}\:{and}\:\boldsymbol{{m}}\:{minutes} \\ $$$$\theta=\left(\frac{{h}+\frac{{m}}{\mathrm{60}}}{\mathrm{12}}−\frac{{m}}{\mathrm{60}}\right)×\mathrm{360}°=\left(\frac{{h}}{\mathrm{12}}−\frac{\mathrm{11}{m}}{\mathrm{720}}\right)×\mathrm{360}° \\ $$$$ \\ $$$${at}\:\mathrm{4}:\mathrm{00} \\ $$$$\Rightarrow\theta=\frac{\mathrm{4}}{\mathrm{12}}×\mathrm{360}=\mathrm{120}° \\ $$$${distance}: \\ $$$${d}=\sqrt{{l}_{{H}} ^{\mathrm{2}} +{l}_{{M}} ^{\mathrm{2}} −\mathrm{2}{l}_{{H}} {l}_{{M}} \mathrm{cos}\:\theta} \\ $$$$\Rightarrow{d}=\sqrt{\mathrm{3}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} −\mathrm{2}×\mathrm{3}×\mathrm{5}×\left(−\mathrm{0}.\mathrm{5}\right)}=\mathrm{7}\:{cm} \\ $$$$ \\ $$$${at}\:\mathrm{4}:\mathrm{40} \\ $$$$\theta=\left(\frac{\mathrm{4}}{\mathrm{12}}−\frac{\mathrm{11}×\mathrm{40}}{\mathrm{720}}\right)×\mathrm{360}°=−\mathrm{100}° \\ $$$$\Rightarrow{d}=\sqrt{\mathrm{3}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} −\mathrm{2}×\mathrm{3}×\mathrm{5}×\mathrm{cos}\:\left(−\mathrm{100}°\right)}=\mathrm{6}.\mathrm{26}\:{cm} \\ $$
Commented by Mr eaay last updated on 16/Dec/17
pls expand more....How do u get the angle...
$${pls}\:{expand}\:{more}….{How}\:{do}\:{u}\:{get}\:{the}\:{angle}… \\ $$
Commented by mrW1 last updated on 16/Dec/17
at 4:00 the hour hand is at position “4”  and the minute hand is at position “0”.  12 hours mean 360°, so 4 hours mean  120°.
$${at}\:\mathrm{4}:\mathrm{00}\:{the}\:{hour}\:{hand}\:{is}\:{at}\:{position}\:“\mathrm{4}'' \\ $$$${and}\:{the}\:{minute}\:{hand}\:{is}\:{at}\:{position}\:“\mathrm{0}''. \\ $$$$\mathrm{12}\:{hours}\:{mean}\:\mathrm{360}°,\:{so}\:\mathrm{4}\:{hours}\:{mean} \\ $$$$\mathrm{120}°. \\ $$

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