Question-208782

Question Number 208782 by cherokeesay last updated on 22/Jun/24

Answered by mr W last updated on 23/Jun/24

Commented by cherokeesay last updated on 23/Jun/24

t h a n k y o u m a s t e r!

Commented by Tawa11 last updated on 23/Jun/24

Weldone sir .

Commented by mr W last updated on 23/Jun/24

\frac{r}{R} = \sqrt{\frac{25}{49}} = \frac{5}{7} \Rightarrow R = \frac{7 r}{5}

d = 2 r

{(R - r)}^{2} + {(\sqrt{15})}^{2} = {(3 r - R)}^{2}

25 = 4 r^{2}

\Rightarrow r = \frac{5}{2} \Rightarrow R = \frac{7}{2}

\sqrt{R^{2} - b^{2}} + \sqrt{r^{2} - b^{2}} = 3 r - R

\sqrt{R^{2} - b^{2}} = 4 - \sqrt{r^{2} - b^{2}}

5 = 4 \sqrt{r^{2} - b^{2}}

b^{2} = r^{2} - \frac{25}{16} \Rightarrow b = \frac{5 \sqrt{3}}{4}

\sin α_{1} = \frac{b}{R} = \frac{5 \sqrt{3}}{14}

\sin α_{2} = \frac{b}{r} = \frac{\sqrt{3}}{2} \Rightarrow α_{2} = \frac{π}{3}

g r e e n a r e a :

\frac{R^{2}}{4} (2 α_{1} - \sin 2 α_{1}) + \frac{r^{2}}{4} (2 α_{2} - \sin 2 α_{2})

= \frac{7^{2}}{4 \times 4} ({2 \sin}^{- 1} \frac{5 \sqrt{3}}{14} - \frac{2 \times 5 \sqrt{3} \times 11}{14^{2}}) + \frac{5^{2}}{4 \times 4} (\frac{2 π}{3} - \frac{\sqrt{3}}{2})

= \frac{49}{8} \sin^{- 1} \frac{5 \sqrt{3}}{14} + \frac{25 π}{24} - \frac{5 \sqrt{3}}{2} \approx 3.027

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