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Question Number 205428 by mr W last updated on 21/Mar/24

Answered by A5T last updated on 21/Mar/24

Commented by A5T last updated on 21/Mar/24

$$BE=\sqrt{s^2+y^2};DF=\frac{s(s-y)}{y}\Rightarrow FE=\frac{(s-y)\sqrt{s^2+y^2}}{y}$$$$FB=\frac{s\sqrt{s^2+y^2}}{y};AF=\frac{s(s-y)}{y}+s=\frac{s^2}{y};\frac{AF\times AB}{2}=\frac{s^3}{2y}$$$$=6(\frac{s^2}{y}+\frac{s\sqrt{s^2+y^2}}{y}+s)\Rightarrow \frac{s^2}{12y}=\frac{s+\sqrt{s^2+y^2}+y}{y}$$$$sy=5(s+y+\sqrt{s^2+y^2})\Rightarrow sy=\frac{5s^2}{12}\Rightarrow y=\frac{5s}{12}$$$$\Rightarrow \frac{5s^2}{12}=5(\frac{17s}{12}+\sqrt{s^2+\frac{25s^2}{144}})=150s\Rightarrow s=30$$$$Areaofwhiteregion=$$$$144+144\pi -\frac{144\pi }{4}+25+25\pi -\frac{25\pi }{4}=169+\frac{507}{4}\pi$$$$\Rightarrow Areaofblueregion=s^2-169-\frac{507\pi }{4}$$$$=900-169-\frac{507\pi }{4}=731-\frac{507\pi }{4}sq.units.$$

Commented by mr W last updated on 21/Mar/24

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Answered by mr W last updated on 21/Mar/24

Commented by mr W last updated on 21/Mar/24

$$\mathrm{\Delta }CEB\backsim \mathrm{\Delta }ABF$$$$\frac{CE}{AB}=\frac{5}{12}\Rightarrow CE=\frac{5a}{12}$$$${\mathrm{\Delta }}_{CEB}=\frac{1}{2}\times a\times \frac{5a}{12}=\frac{1}{2}\times (a+\frac{5a}{12}+\sqrt{a^2+{\left(\frac{5a}{12}\right)}^2})\times 5$$$$\Rightarrow a=12+5+\sqrt{{12}^2+5^2}=30$$$$shadedarea={30}^2-({12}^2+5^2)-\frac{3\pi }{4}({12}^2+5^2)$$$$=731-\frac{507\pi }{4}\approx 332.8$$

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