Question-211537

Question Number 211537 by cherokeesay last updated on 12/Sep/24

Answered by mr W last updated on 12/Sep/24

(√((R−2)^2 −2^2 ))−2=(√(R^2 −6^2 )) (√(R^2 −4R))=2+(√(R^2 −6^2 )) 8−R=(√(R^2 −6^2 )) ⇒R=((50)/8) ratio=(2^2 /(0.5R^2 ))=((128)/(625))=20.48%

$$\sqrt{\left({R}−\mathrm{2}\right)^{\mathrm{2}} −\mathrm{2}^{\mathrm{2}} }−\mathrm{2}=\sqrt{{R}^{\mathrm{2}} −\mathrm{6}^{\mathrm{2}} } \\ $$$$\sqrt{{R}^{\mathrm{2}} −\mathrm{4}{R}}=\mathrm{2}+\sqrt{{R}^{\mathrm{2}} −\mathrm{6}^{\mathrm{2}} } \\ $$$$\mathrm{8}−{R}=\sqrt{{R}^{\mathrm{2}} −\mathrm{6}^{\mathrm{2}} } \\ $$$$\Rightarrow{R}=\frac{\mathrm{50}}{\mathrm{8}} \\ $$$${ratio}=\frac{\mathrm{2}^{\mathrm{2}} }{\mathrm{0}.\mathrm{5}{R}^{\mathrm{2}} }=\frac{\mathrm{128}}{\mathrm{625}}=\mathrm{20}.\mathrm{48\%} \\ $$

Commented by mr W last updated on 12/Sep/24

$${thanks}! \\ $$

Answered by A5T last updated on 12/Sep/24

6^2 =x(2R−x)⇒2R−x=((36)/x)...(i) (R−2)^2 =2^2 +(2+R−x)^2 ⇒2R−x=(4/(x−4))...(ii) (i)&(ii)⇒(4/(x−4))=((36)/x)⇒x=9x−36⇒x=(9/2)⇒R=((25)/4) ⇒(([green])/([white]))=(4/((625)/(32)))=((128)/(625))=0.2048

$$\mathrm{6}^{\mathrm{2}} ={x}\left(\mathrm{2}{R}−{x}\right)\Rightarrow\mathrm{2}{R}−{x}=\frac{\mathrm{36}}{{x}}…\left({i}\right) \\ $$$$\left({R}−\mathrm{2}\right)^{\mathrm{2}} =\mathrm{2}^{\mathrm{2}} +\left(\mathrm{2}+{R}−{x}\right)^{\mathrm{2}} \Rightarrow\mathrm{2}{R}−{x}=\frac{\mathrm{4}}{{x}−\mathrm{4}}…\left({ii}\right) \\ $$$$\left({i}\right)\&\left({ii}\right)\Rightarrow\frac{\mathrm{4}}{{x}−\mathrm{4}}=\frac{\mathrm{36}}{{x}}\Rightarrow{x}=\mathrm{9}{x}−\mathrm{36}\Rightarrow{x}=\frac{\mathrm{9}}{\mathrm{2}}\Rightarrow{R}=\frac{\mathrm{25}}{\mathrm{4}} \\ $$$$\Rightarrow\frac{\left[{green}\right]}{\left[{white}\right]}=\frac{\mathrm{4}}{\frac{\mathrm{625}}{\mathrm{32}}}=\frac{\mathrm{128}}{\mathrm{625}}=\mathrm{0}.\mathrm{2048} \\ $$

Commented by A5T last updated on 12/Sep/24