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Question Number 214603 by ajfour last updated on 13/Dec/24

Answered by TonyCWX08 last updated on 13/Dec/24

$$WhereisR?$$$$Whereisr?$$

Commented by ajfour last updated on 13/Dec/24

$$sorrywrongquestion$$

Answered by mr W last updated on 13/Dec/24

Commented by mr W last updated on 13/Dec/24

$${(R-r)}^2=R^2+{\left(2r\right)}^2-4Rr\cos 30°$$$$2(\sqrt{3}-1)R=3r$$$$\Rightarrow \frac{R}{r}=\frac{3}{2(\sqrt{3}-1)}=\frac{3(\sqrt{3}+1)}{4}\approx 2.049$$

Answered by A5T last updated on 13/Dec/24

$$\sqrt{{(R-r)}^2-r^2}=R-\sqrt{{\left(2r\right)}^2-r^2}$$$$\Rightarrow \sqrt{R(R-2r)}=R-r\sqrt{3}$$$$\Rightarrow R^2-2rR=R^2+3r^2-2Rr\sqrt{3}$$$$\Rightarrow 2Rr(\sqrt{3}-1)=3r^2\Rightarrow \frac{R}{r}=\frac{3}{2(\sqrt{3}-1)}=\frac{3(\sqrt{3}+1)}{4}$$

Commented by ajfour last updated on 13/Dec/24

$$Thankyou!Promptways!$$

Commented by ajfour last updated on 13/Dec/24

https://youtu.be/z1RwkyYqV34?si=luatRZ1-3shsgmnX

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