Question-179386

Question Number 179386 by cherokeesay last updated on 29/Oct/22

Answered by DvMc last updated on 29/Oct/22

(y/r)=(3/4) y=(3/4)r (x/r)=(4/3) x=(4/3)r Sum of areas (1/2)yr+r^2 +(1/2)rx=((3×4)/2) ((49)/(24))r^2 =6 r=((12)/7) cos(α)=(4/5)=((h1)/r) h1=((48)/(35)) For yellow triangle A=((bh)/2) where b=y h=r+h1 A=(1/2)×(3/4)×((12)/7)×(((12)/7)+((48)/(35)))=((486)/(245))

$$\frac{{y}}{{r}}=\frac{\mathrm{3}}{\mathrm{4}}\:\:\:\:\:\:\:\:\:\:\:{y}=\frac{\mathrm{3}}{\mathrm{4}}{r} \\ $$$$\frac{{x}}{{r}}=\frac{\mathrm{4}}{\mathrm{3}}\:\:\:\:\:\:\:\:\:\:\:{x}=\frac{\mathrm{4}}{\mathrm{3}}{r} \\ $$$${Sum}\:{of}\:{areas} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{yr}+{r}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{2}}{rx}=\frac{\mathrm{3}×\mathrm{4}}{\mathrm{2}} \\ $$$$\frac{\mathrm{49}}{\mathrm{24}}{r}^{\mathrm{2}} =\mathrm{6}\:\:\:\:\:\:\:\:\:\:\:{r}=\frac{\mathrm{12}}{\mathrm{7}} \\ $$$${cos}\left(\alpha\right)=\frac{\mathrm{4}}{\mathrm{5}}=\frac{{h}\mathrm{1}}{{r}}\:\:\:\:\:\:{h}\mathrm{1}=\frac{\mathrm{48}}{\mathrm{35}} \\ $$$${For}\:{yellow}\:{triangle} \\ $$$${A}=\frac{{bh}}{\mathrm{2}}\:\:\:\:\:{where} \\ $$$${b}={y}\:\:\:\:\:\:\:{h}={r}+{h}\mathrm{1} \\ $$$${A}=\frac{\mathrm{1}}{\mathrm{2}}×\frac{\mathrm{3}}{\mathrm{4}}×\frac{\mathrm{12}}{\mathrm{7}}×\left(\frac{\mathrm{12}}{\mathrm{7}}+\frac{\mathrm{48}}{\mathrm{35}}\right)=\frac{\mathrm{486}}{\mathrm{245}} \\ $$$$ \\ $$

Commented by DvMc last updated on 29/Oct/22

Commented by Acem last updated on 29/Oct/22

$${Hi},\:{please}\:{how}\:{to}\:{upload}\:{a}\:{pic}.\:{on}\:{comments}? \\ $$

Commented by DvMc last updated on 29/Oct/22

$${Hi}.\:{Ijust}\:{pick}\:{the}\:{option}.\:{Maybe}\:{you}\:{need} \\ $$$${upgrade}\:{your}\:{app}\:{to}\:{latest}\:{version} \\ $$$$ \\ $$

Commented by DvMc last updated on 29/Oct/22

Commented by Acem last updated on 29/Oct/22

Commented by cherokeesay last updated on 29/Oct/22

$${Nice}\:!\:{thank}\:{you}\:{so}\:{much}. \\ $$

Commented by cherokeesay last updated on 29/Oct/22

$${Nice}\:!\:{Thank}\:{you}\:{sir} \\ $$

Answered by mr W last updated on 29/Oct/22

((3−r)/r)=(r/(4−r)) ⇒r=((12)/7) ((4−r)/4)×5−r=(8/7) 5−(8/7)=((27)/7) yellow area =((3−((12)/7))/3)×(((27)/7)/5)×((3×4)/2)=((486)/(245))

$$\frac{\mathrm{3}−{r}}{{r}}=\frac{{r}}{\mathrm{4}−{r}} \\ $$$$\Rightarrow{r}=\frac{\mathrm{12}}{\mathrm{7}} \\ $$$$\frac{\mathrm{4}−{r}}{\mathrm{4}}×\mathrm{5}−{r}=\frac{\mathrm{8}}{\mathrm{7}} \\ $$$$\mathrm{5}−\frac{\mathrm{8}}{\mathrm{7}}=\frac{\mathrm{27}}{\mathrm{7}} \\ $$$${yellow}\:{area}\:=\frac{\mathrm{3}−\frac{\mathrm{12}}{\mathrm{7}}}{\mathrm{3}}×\frac{\frac{\mathrm{27}}{\mathrm{7}}}{\mathrm{5}}×\frac{\mathrm{3}×\mathrm{4}}{\mathrm{2}}=\frac{\mathrm{486}}{\mathrm{245}} \\ $$

Commented by cherokeesay last updated on 29/Oct/22

$${thank}\:{you}\:{master}\:! \\ $$

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