Question-177586

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Question Number 177586 by aurpeyz last updated on 07/Oct/22

Commented by aurpeyz last updated on 07/Oct/22

$${pls}\:{help}.\:{the}\:{question}\:{is}\:{giving}\:{me}\:{headache} \\ $$

Commented by mr W last updated on 07/Oct/22

$${the}\:{quality}\:{of}\:{the}\:{image}\:{is}\:{giving}\:{me} \\ $$$${headache}\::\left\{\right) \\ $$

Commented by Ar Brandon last updated on 07/Oct/22

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Commented by aurpeyz last updated on 07/Oct/22

$${thank}\:{you}\:{for}\:{solving}\:{it}\: \\ $$

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

$${this}\:{is}\:{what}\:{they}\:{chose}\:{as}\:{the}\:{answer} \\ $$

Commented by aurpeyz last updated on 07/Oct/22

$${is}\:{the}\:{length}\:{of}\:{line}\:{segment}\:{AB}\:{the}\:{same}\:{as}\:{length}\:{of}\:{arc}\:{in}\:{this}\:{case}? \\ $$

Commented by mr W last updated on 07/Oct/22

$${certainly}\:{no}! \\ $$$$\overset{\frown} {{AB}}\:>\:{AB}\:\:{always}! \\ $$

Commented by mr W last updated on 07/Oct/22

$${they}\:{from}\:{the}\:{book}\:{are}\:{wrong}!\:{since} \\ $$$${in}\:{the}\:{question}\:{quantity}\:{A}\:{is}\:{the}\:{line} \\ $$$${segment}\:{AB},\:{very}\:{clear}! \\ $$

Commented by aurpeyz last updated on 07/Oct/22