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Question Number 203392 by professorleiciano last updated on 18/Jan/24

Commented by a.lgnaoui last updated on 19/Jan/24

Commented by a.lgnaoui last updated on 20/Jan/24

$$△\mathbf{OAB}{\mathbf{a}}^2=41-40\cos \mathbf{x}$$$$\mathbf{a}=\sqrt{41-40\cos \mathbf{x}}$$$$\{\begin{array}{l}4\sin \mathbf{x}=\mathbf{a}\sin \mathbf{y}\\ 4\cos \mathbf{x}+\mathbf{a}\cos \mathbf{y}=5\end{array}$$$$\mathbf{Aire}\left(\mathbf{OAB}\right)=\frac{1}{2}\left(4\times 5\right)\sin \mathbf{x}=10\sin \mathbf{x}$$$$\mathbf{Aire}\left(\mathbf{OAD}\right)=\frac{1}{2}\mathbf{OC}\times 3\sin \mathbf{x}$$$$=\frac{3}{2}\times \frac{12}{5}\sin \mathbf{x}=\frac{18}{5}\sin \mathbf{x}$$$$$$$$\mathbf{Aire}[\left(\mathbf{OAB}\right)+\left(\mathbf{OCD}\right)]=\frac{68}{5}\sin \mathbf{x}\left(1\right)$$$$$$$$$$$$\mathbf{Calcul}\mathbf{de}\mathbf{x}$$$$\mathbf{h}=(4+\frac{12}{5})\sin \mathbf{x}=\frac{32}{5}\sin \mathbf{x}$$$$\mathbf{Aire}\mathbf{totale}=\frac{(3+5)\mathbf{h}}{2}=4\mathbf{h}$$$$=\mathbf{Aird}\left(\mathbf{BCD}\right)+\mathbf{ACD}=$$$$\mathbf{Aire}=2\mathbf{A}\mathrm{ird}\left(\mathbf{OCD}\right)+\mathbf{A}2+\mathbf{A}3$$$$=\mathbf{Aird}\mathbf{T}\mathrm{itale}-\mathbf{OAB}+\mathbf{OCD}$$$$$$$$\frac{3\mathrm{ACsin}\mathbf{x}}{2}+\frac{3\mathbf{BD}\sin \mathbf{y}}{2}=$$$$\frac{128\sin \mathbf{x}}{5}-\mathbf{Aire}\left(\mathbf{OAB}\right)+\mathbf{Aire}\left(\mathbf{OCD}\right)$$$$\Rightarrow \mathbf{Aire}\left(\mathbf{OAB}\right)-\mathbf{Aire}\left(\mathbf{OCD}\right)\left(2\right)$$$$=\frac{128\sin \mathbf{x}}{5}-\frac{3}{2}(\mathbf{AC}\sin \mathbf{x}+\mathbf{BD}\sin \mathbf{y})$$$$\frac{\mathbf{OD}}{\mathbf{BD}}=\frac{12/5}{4+12/5}=\frac{12}{32}=\frac{3}{8}\mathbf{BD}=\frac{8}{3}\mathbf{OD}$$$$\frac{\mathbf{OD}}{\mathbf{OB}}=\frac{3}{5}\mathbf{OD}=\frac{3}{5}\mathbf{a}$$$$\{\begin{array}{l}\mathbf{AC}=4+\frac{12}{5}=\frac{32}{5}\\ \mathbf{BD}=\frac{8}{3}\times \frac{3}{5}\mathbf{a}=\frac{8}{5}\mathbf{a}\end{array}$$$$$$$$\frac{\mathbf{OAB}}{\mathbf{A}\mathrm{O}\mathbf{D}}=\frac{25}{9}\frac{\mathbf{OAB}+\mathbf{A}\mathrm{O}\mathbf{D}}{\mathbf{OAB}}=\frac{34}{25}\left(3\right)$$$$$$$$△\mathbf{ABC}\mathbf{AB}\sin \mathit{\lambda }=\mathbf{AC}\sin \mathbf{x}$$$$\mathbf{b}\sin \mathit{\lambda }=\frac{32}{5}\sin \mathbf{x}$$$$\mathbf{b}=\sqrt{25+{\left(\frac{32}{5}\right)}^2-64\cos \mathbf{x}}$$$$$$$$\frac{32}{5}\sin \mathbf{x}=\sqrt{\frac{1649}{25}-64\cos \mathbf{x}}$$$$$$$${32}^2\cos {}^2\mathrm{x}-1600\cos \mathrm{x}+625=0$$$$\cos \mathrm{x}=\frac{25}{32}\sin \mathrm{x}=\frac{\sqrt{399}}{32}\cong \frac{20}{32}=\frac{5}{8}$$$$\left(1\right)\Rightarrow$$$$\mathbf{Aire}[\left(\mathrm{AOB}\right)+\left(\mathbf{A}\mathrm{OD}\right)]=\frac{68}{5}\sin \mathbf{x}=\frac{17}{2}{\mathrm{cm}}^2$$$$\mathbf{donc}:\mathbf{R}\mathrm{eponse}=\mathbf{e}$$

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