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Question Number 42866 by Cheyboy last updated on 03/Sep/18

Answered by MJS last updated on 04/Sep/18

$$x\to r\mathrm{to}\mathrm{not}\mathrm{get}\mathrm{confused}$$$$\mathrm{small}\mathrm{circle}\mathrm{touching}\mathrm{half}\mathrm{circle}\mathrm{from}\mathrm{inside}$$$$r^2+p^2-{(\frac{R}{2}-r)}^2=0\Rightarrow p^2-\frac{R^2}{4}+Rr=0$$$$\mathrm{small}\mathrm{circle}\mathrm{touching}\sim \mathrm{quarter}\mathrm{circle}\mathrm{from}\mathrm{outside}$$$$r^2+{(\frac{3R}{2}+p)}^2-{(\frac{3R}{2}+r)}^2=0\Rightarrow 3Rp+p^2-3Rr=0$$$$$$$$\left(1\right)Rr+p^2-\frac{R^2}{4}=0$$$$\left(2\right)-3Rr+p^2+3Rp=0$$$$\left(1\right)-\left(2\right)4Rr-3Rp-\frac{R^2}{4}=0$$$$4r-3p-\frac{R}{4}=0\Rightarrow p=\frac{4}{3}r-\frac{R}{12}$$$$\left(1\right)Rr+{(\frac{4}{3}r-\frac{R}{12})}^3-\frac{R^2}{4}=0$$$$r^2+\frac{7R}{16}r-\frac{35R^2}{256}=0$$$$r=\frac{-7+3\sqrt{21}}{32}R$$$$p=\frac{-3+\sqrt{21}}{8}R$$$$R=8\Rightarrow p=1.58r=1.69$$$$$$$$[\mathrm{btw}\mathrm{the}{2}^{\mathrm{nd}}\mathrm{solution}\mathrm{for}r\mathrm{and}p\mathrm{gives}\mathrm{a}\mathrm{circle}$$$$\mathrm{touching}\mathrm{the}\mathrm{half}\mathrm{circle}\mathrm{from}\mathrm{outside}\mathrm{and}$$$$\mathrm{the}\sim \mathrm{quarter}\mathrm{circle}\mathrm{from}\mathrm{inside}]$$

Commented by Cheyboy last updated on 04/Sep/18

$$Thankyousir$$

Commented by MJS last updated on 04/Sep/18

$$\mathrm{made}\mathrm{a}\mathrm{strange}\mathrm{mistake},\mathrm{corrected}\mathrm{now}$$

Commented by Cheyboy last updated on 04/Sep/18

$$oksirthankyour$$

Answered by MrW3 last updated on 04/Sep/18

$${(\sqrt{{(\frac{R}{2}-x)}^2-x^2}+\frac{3R}{2})}^2+x^2={(\frac{3R}{2}+x)}^2$$$${(\frac{R}{2}-x)}^2-x^2+2\left(\frac{3R}{2}\right)\sqrt{{(\frac{R}{2}-x)}^2-x^2}+\frac{9R^2}{4}+x^2=\frac{9R^2}{4}+3Rx+x^2$$$$\frac{R^2}{4}-Rx+x^2-x^2+3R\sqrt{{(\frac{R}{2}-x)}^2-x^2}+\frac{9R^2}{4}+x^2=\frac{9R^2}{4}+3Rx+x^2$$$$3\sqrt{{(\frac{R}{2}-x)}^2-x^2}=4x-\frac{R}{4}$$$$9(\frac{R^2}{4}-Rx+x^2-x^2)=16x^2-2Rx+\frac{R^2}{16}$$$$16x^2+7Rx-\frac{35R^2}{16}=0$$$$\frac{x}{R}=\frac{-7+\sqrt{49+4\times 35}}{2\times 16}=\frac{\sqrt{189}-7}{32}\approx 0.21$$$$\Rightarrow x\approx 0.21\times 8=1.69$$

Commented by MrW3 last updated on 04/Sep/18

Commented by Cheyboy last updated on 04/Sep/18

$$Waawmybossthankalotur$$$$diagramisvryhelpful$$

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