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
Commented by aurpeyz last updated on 07/Oct/22
$${thank}\:{you}\:{for}\:{solving}\:{it}\: \\ $$
Answered by mr W last updated on 07/Oct/22
$${quantity}\:{A}\:{is}\:{the}\:{line}\:{segment}\:{AB}. \\ $$$${AB}=\mathrm{2}{r}×\mathrm{sin}\:\frac{\angle{AOB}}{\mathrm{2}} \\ $$$$\mathrm{0}<\angle{AOB}<\frac{\pi}{\mathrm{2}} \\ $$$$\mathrm{0}<\frac{\angle{AOB}}{\mathrm{2}}<\frac{\pi}{\mathrm{4}} \\ $$$$\mathrm{0}<\mathrm{sin}\:\frac{\angle{AOB}}{\mathrm{2}}<\frac{\sqrt{\mathrm{2}}}{\mathrm{2}} \\ $$$$\mathrm{0}<\mathrm{2}{r}\:\mathrm{sin}\:\frac{\angle{AOB}}{\mathrm{2}}<\sqrt{\mathrm{2}}{r}<\mathrm{1}.\mathrm{5}{r} \\ $$$$\Rightarrow{quantity}\:{B}\:{is}\:{greater}\:{than}\:{A}. \\ $$
Commented by mr W last updated on 07/Oct/22
$${if}\:{quantity}\:{A}\:{is}\:{the}\:{arc}\:{segment}\:\overset{\frown} {{AB}}. \\ $$$$\overset{\frown} {{AB}}=\angle{AOB}_{{in}\:{rad}} ×{r} \\ $$$$\mathrm{0}<\angle{AOB}<\frac{\pi}{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{0}<\overset{\frown} {{AB}}<\frac{\pi}{\mathrm{2}}{r}\approx\mathrm{1}.\mathrm{57}{r} \\ $$$${quantity}\:{A}\:{is}\:{between}\:\mathrm{0}\:{and}\:\mathrm{1}.\mathrm{57}{r}. \\ $$$${quantity}\:{B}\:{is}\:\mathrm{1}.\mathrm{5}{r}. \\ $$$${that}\:{means}\:{with}\:{given}\:{information} \\ $$$${you}\:{can}\:{not}\:{say}\:{which}\:{is}\:{greater}. \\ $$
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
$${so}\:{the}\:{right}\:{answer}\:{is}\:{B}.\:{your}\:{first}\:{solution}? \\ $$
Commented by mr W last updated on 07/Oct/22
$${yes}. \\ $$
Commented by aurpeyz last updated on 07/Oct/22
$${thanks} \\ $$