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Question-103062




Question Number 103062 by ajfour last updated on 12/Jul/20
Commented by ajfour last updated on 12/Jul/20
If both coloured regions have  equal areas, find radius of the  semicircle.
$${If}\:{both}\:{coloured}\:{regions}\:{have} \\ $$$${equal}\:{areas},\:{find}\:{radius}\:{of}\:{the} \\ $$$${semicircle}. \\ $$
Answered by mr W last updated on 12/Jul/20
area of segment:  (R^2 /2)(2θ−sin 2θ)=((πR^2 )/4)  2θ−sin 2θ=(π/2)  ⇒θ=1.1549 (=66.17°)    tan θ=y′=2R sin θ  ⇒R=(1/(2 cos θ))≈1.2376
$${area}\:{of}\:{segment}: \\ $$$$\frac{{R}^{\mathrm{2}} }{\mathrm{2}}\left(\mathrm{2}\theta−\mathrm{sin}\:\mathrm{2}\theta\right)=\frac{\pi{R}^{\mathrm{2}} }{\mathrm{4}} \\ $$$$\mathrm{2}\theta−\mathrm{sin}\:\mathrm{2}\theta=\frac{\pi}{\mathrm{2}} \\ $$$$\Rightarrow\theta=\mathrm{1}.\mathrm{1549}\:\left(=\mathrm{66}.\mathrm{17}°\right) \\ $$$$ \\ $$$$\mathrm{tan}\:\theta={y}'=\mathrm{2}{R}\:\mathrm{sin}\:\theta \\ $$$$\Rightarrow{R}=\frac{\mathrm{1}}{\mathrm{2}\:\mathrm{cos}\:\theta}\approx\mathrm{1}.\mathrm{2376} \\ $$
Commented by mr W last updated on 12/Jul/20
Commented by ajfour last updated on 12/Jul/20
Short, succinct and Nice Sir!
$${Short},\:{succinct}\:{and}\:{Nice}\:{Sir}! \\ $$

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