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Question Number 63985 by Tawa1 last updated on 11/Jul/19

Answered by mr W last updated on 12/Jul/19

Commented by mr W last updated on 12/Jul/19

$$\tan \alpha =\frac{1}{2}$$$$\theta =\frac{\pi }{4}-\alpha$$$$radiysR=3$$$$BC=\sqrt{2}R=3\sqrt{2}$$$$CE=2R\cos \alpha =2\times 3\times \frac{2}{\sqrt{5}}=\frac{12}{\sqrt{5}}$$$$A_{\mathrm{\Delta }ABC}=\frac{3\times 3}{2}=\frac{9}{2}$$$$A_{\mathrm{\Delta }BCE}=\frac{BC\times EC\sin \theta }{2}=\frac{3\sqrt{2}\times 12}{2\sqrt{5}}\times \frac{\sqrt{2}}{2}(\frac{2}{\sqrt{5}}-\frac{1}{\sqrt{5}})=\frac{18}{5}$$$$A_{capBE}=\frac{3^2}{2}(2\theta -\sin 2\theta )=\frac{9}{2}(\frac{\pi }{2}-2\alpha -\cos 2\alpha )$$$$A_{capBE}=\frac{9}{2}(\frac{\pi }{2}-2\alpha -1+2\times \frac{1}{5})$$$$A_{capBE}=\frac{9}{2}(\frac{\pi }{2}-{2\tan }^{-1}\frac{1}{2}-\frac{3}{5})$$$$A_{green}=A_{\mathrm{\Delta }ABC}-A_{\mathrm{\Delta }BCE}-A_{capBE}$$$$=\frac{9}{2}-\frac{18}{5}-\frac{9}{2}(\frac{\pi }{2}-{2\tan }^{-1}\frac{1}{2}-\frac{3}{5})$$$$=\frac{9}{2}(\frac{4}{5}-\frac{\pi }{2}+{2\tan }^{-1}\frac{1}{2})$$$$A_{blue}=6\times 12-2\times \pi \times 3^2-2\times \frac{9}{2}(\frac{4}{5}-\frac{\pi }{2}+{2\tan }^{-1}\frac{1}{2})$$$$A_{blue}=\frac{324}{5}-\frac{27\pi }{2}-18{\tan }^{-1}\frac{1}{2}=14.04$$

Commented by Tawa1 last updated on 12/Jul/19

$$\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}$$

Commented by Tawa1 last updated on 12/Jul/19

$$\mathrm{I}\mathrm{appreciate}\mathrm{your}\mathrm{effort}$$

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