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Question-47515




Question Number 47515 by ajfour last updated on 11/Nov/18
Commented by ajfour last updated on 11/Nov/18
Find α in terms of R if both  coloured areas are equal.
$${Find}\:\alpha\:{in}\:{terms}\:{of}\:{R}\:{if}\:{both} \\ $$$${coloured}\:{areas}\:{are}\:{equal}. \\ $$
Answered by mr W last updated on 11/Nov/18
A_(blue) =R^2  cot (α/2)−(((π−α)R^2 )/2)=A_(yellow) =πR^2   ⇒cot (α/2)+(α/2)=((3π)/2)  ⇒α=25.0932°
$${A}_{{blue}} ={R}^{\mathrm{2}} \:\mathrm{cot}\:\frac{\alpha}{\mathrm{2}}−\frac{\left(\pi−\alpha\right){R}^{\mathrm{2}} }{\mathrm{2}}={A}_{{yellow}} =\pi{R}^{\mathrm{2}} \\ $$$$\Rightarrow\mathrm{cot}\:\frac{\alpha}{\mathrm{2}}+\frac{\alpha}{\mathrm{2}}=\frac{\mathrm{3}\pi}{\mathrm{2}} \\ $$$$\Rightarrow\alpha=\mathrm{25}.\mathrm{0932}° \\ $$
Commented by ajfour last updated on 11/Nov/18
Thanks Sir, very quick!
$${Thanks}\:{Sir},\:{very}\:{quick}! \\ $$

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