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Question-177586




Question Number 177586 by aurpeyz last updated on 07/Oct/22
Commented by aurpeyz last updated on 07/Oct/22
pls help. the question is giving me headache
$${pls}\:{help}.\:{the}\:{question}\:{is}\:{giving}\:{me}\:{headache} \\ $$
Commented by mr W last updated on 07/Oct/22
the quality of the image is giving me  headache :{)
$${the}\:{quality}\:{of}\:{the}\:{image}\:{is}\:{giving}\:{me} \\ $$$${headache}\::\left\{\right) \\ $$
Commented by Ar Brandon last updated on 07/Oct/22
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Commented by aurpeyz last updated on 07/Oct/22
thank you for solving it
$${thank}\:{you}\:{for}\:{solving}\:{it}\: \\ $$
Answered by mr W last updated on 07/Oct/22
quantity A is the line segment AB.  AB=2r×sin ((∠AOB)/2)  0<∠AOB<(π/2)  0<((∠AOB)/2)<(π/4)  0<sin ((∠AOB)/2)<((√2)/2)  0<2r sin ((∠AOB)/2)<(√2)r<1.5r  ⇒quantity B is greater than A.
$${quantity}\:{A}\:{is}\:{the}\:{line}\:{segment}\:{AB}. \\ $$$${AB}=\mathrm{2}{r}×\mathrm{sin}\:\frac{\angle{AOB}}{\mathrm{2}} \\ $$$$\mathrm{0}<\angle{AOB}<\frac{\pi}{\mathrm{2}} \\ $$$$\mathrm{0}<\frac{\angle{AOB}}{\mathrm{2}}<\frac{\pi}{\mathrm{4}} \\ $$$$\mathrm{0}<\mathrm{sin}\:\frac{\angle{AOB}}{\mathrm{2}}<\frac{\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$\mathrm{0}<\mathrm{2}{r}\:\mathrm{sin}\:\frac{\angle{AOB}}{\mathrm{2}}<\sqrt{\mathrm{2}}{r}<\mathrm{1}.\mathrm{5}{r} \\ $$$$\Rightarrow{quantity}\:{B}\:{is}\:{greater}\:{than}\:{A}. \\ $$
Commented by mr W last updated on 07/Oct/22
if quantity A is the arc segment AB^(⌢) .  AB^(⌢) =∠AOB_(in rad) ×r  0<∠AOB<(π/2)  ⇒0<AB^(⌢) <(π/2)r≈1.57r  quantity A is between 0 and 1.57r.  quantity B is 1.5r.  that means with given information  you can not say which is greater.
$${if}\:{quantity}\:{A}\:{is}\:{the}\:{arc}\:{segment}\:\overset{\frown} {{AB}}. \\ $$$$\overset{\frown} {{AB}}=\angle{AOB}_{{in}\:{rad}} ×{r} \\ $$$$\mathrm{0}<\angle{AOB}<\frac{\pi}{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{0}<\overset{\frown} {{AB}}<\frac{\pi}{\mathrm{2}}{r}\approx\mathrm{1}.\mathrm{57}{r} \\ $$$${quantity}\:{A}\:{is}\:{between}\:\mathrm{0}\:{and}\:\mathrm{1}.\mathrm{57}{r}. \\ $$$${quantity}\:{B}\:{is}\:\mathrm{1}.\mathrm{5}{r}. \\ $$$${that}\:{means}\:{with}\:{given}\:{information} \\ $$$${you}\:{can}\:{not}\:{say}\:{which}\:{is}\:{greater}. \\ $$
Commented by aurpeyz last updated on 07/Oct/22
this is what they chose as the answer
$${this}\:{is}\:{what}\:{they}\:{chose}\:{as}\:{the}\:{answer} \\ $$
Commented by aurpeyz last updated on 07/Oct/22
is the length of line segment AB the same as length of arc in this case?
$${is}\:{the}\:{length}\:{of}\:{line}\:{segment}\:{AB}\:{the}\:{same}\:{as}\:{length}\:{of}\:{arc}\:{in}\:{this}\:{case}? \\ $$
Commented by mr W last updated on 07/Oct/22
certainly no!  AB^(⌢)  > AB  always!
$${certainly}\:{no}! \\ $$$$\overset{\frown} {{AB}}\:>\:{AB}\:\:{always}! \\ $$
Commented by mr W last updated on 07/Oct/22
they from the book are wrong! since  in the question quantity A is the line  segment AB, very clear!
$${they}\:{from}\:{the}\:{book}\:{are}\:{wrong}!\:{since} \\ $$$${in}\:{the}\:{question}\:{quantity}\:{A}\:{is}\:{the}\:{line} \\ $$$${segment}\:{AB},\:{very}\:{clear}! \\ $$
Commented by aurpeyz last updated on 07/Oct/22
so the right answer is B. your first solution?
$${so}\:{the}\:{right}\:{answer}\:{is}\:{B}.\:{your}\:{first}\:{solution}? \\ $$
Commented by mr W last updated on 07/Oct/22
yes.
$${yes}. \\ $$
Commented by aurpeyz last updated on 07/Oct/22
thanks
$${thanks} \\ $$

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