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Question Number 154736 by mathdanisur last updated on 21/Sep/21

Commented by Tawa11 last updated on 21/Sep/21

$$\mathrm{Weldone}\mathrm{sir}.$$

Answered by mr W last updated on 21/Sep/21

Commented by mr W last updated on 21/Sep/21

$$\frac{16}{\tan \theta }+16+7+\frac{7}{\tan (\frac{\pi }{2}-\frac{\theta }{2})}=\frac{16}{\sin \theta }+\frac{16}{\cos \theta }$$$$23+7\tan \frac{\theta }{2}=16(\frac{1}{\sin \theta }+\frac{1}{\cos \theta }-\frac{1}{\tan \theta })$$$$lett=\tan \frac{\theta }{2}$$$$23+7t=16(\frac{1+t^2}{2t}+\frac{1+t^2}{1-t^2}-\frac{1-t^2}{2t})$$$$23-9t=\frac{16(1+t^2)}{1-t^2}$$$$9t^3-39t^2-9t+7=0$$$$(3t-1)(3t^2-12t-7)=0$$$$t=\frac{1}{3}✓$$$$t=2\pm \frac{\sqrt{51}}{3}(rejected,since0<\tan \frac{\theta }{2}<1)$$$$\tan \frac{\theta }{2}=\frac{1}{3}$$$$\tan \theta =\frac{2\times \frac{1}{3}}{1-{\left(\frac{1}{3}\right)}^2}=\frac{3}{4}$$$$A_{blue}=\frac{16}{2}\times 16\tan \theta +\frac{7}{2}\times \frac{7}{\tan \frac{\theta }{2}}$$$$A_{blue}=128\times \frac{3}{4}+\frac{49}{2\times \frac{1}{3}}$$$$A_{blue}=\frac{339}{2}=169.5$$

Commented by mathdanisur last updated on 21/Sep/21

$$\mathrm{Creativ}\mathrm{solution}\mathbf{S}\mathrm{er},\mathrm{thank}\mathrm{you}$$

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