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Question Number 28574 by ajfour last updated on 27/Jan/18

Commented by ajfour last updated on 27/Jan/18

$F i n d e n t i r e g r e e n a r e a i n t e r m s$ $o f r, θ . T h e t w o f r a m e s a r e$ $s q u a r e f r a m e s w i t h c o m m o n$ $c e n t r e . F i n d a l s o t h e m a x i m u m$ $o f t h e a r e a f o r c o n s t a n t r .$
Commented by ajfour last updated on 29/Jan/18

$m r W 2 S i r, w o u l d y o u s o l v e t h i s ?$
	Answered by mrW2 last updated on 29/Jan/18

	Commented by ajfour last updated on 29/Jan/18

	$t h a n k y o u e n o r m o u s l y S i r!$ $e x t r e m e l y s a t i s f a c t o r y!$
	Commented by mrW2 last updated on 29/Jan/18

	$E q n . o f A B :$ $x = \frac{a}{2} w i t h a = \sqrt{2} r$ $E q n . o f C D (= A B r o t a t e d b y θ)$ $x \cos θ + y \sin θ = \frac{a}{2}$ $\frac{a}{2} \cos θ + y_{E} \sin θ = \frac{a}{2}$ $\Rightarrow y_{E} = \frac{a (1 - \cos θ)}{2 \sin θ}$ $x_{F} \cos θ + \frac{a}{2} \sin θ = \frac{a}{2}$ $\Rightarrow x_{F} = \frac{a (1 - \sin θ)}{2 \cos θ}$ $A_{Δ B E F} = \frac{1}{2} [\frac{a}{2} - \frac{a (1 - \sin θ)}{2 \cos θ}] [\frac{a}{2} - \frac{a (1 - \cos θ)}{2 \sin θ}]$ $A_{Δ B E F} = \frac{a^{2}}{8} [\frac{\cos θ - 1 + \sin θ}{\cos θ}] [\frac{\sin θ - 1 + \cos θ}{\sin θ}]$ $A_{Δ B E F} = \frac{a^{2}}{8} \times \frac{{(\cos θ + \sin θ - 1)}^{2}}{\sin θ \cos θ}$ $A_{Δ B E F} = \frac{a^{2}}{4} [1 - \frac{\cos θ + \sin θ - 1}{\sin θ \cos θ}]$ $A_{G r e e n} = 4 A_{Δ B E F}$ $A_{G r e e n} (θ) = a^{2} [1 - \frac{\cos θ + \sin θ - 1}{\sin θ \cos θ}]$ $A_{G r e e n} (θ) = 2 r^{2} [1 - \frac{\cos θ + \sin θ - 1}{\sin θ \cos θ}]$ $w i t h 0 ⩽ θ ⩽ \frac{π}{2}$ $A_{G r e e n} (\frac{π}{2} + θ) = A_{G r e e n} (θ)$ $f (θ) = 1 - \frac{\cos θ + \sin θ - 1}{\sin θ \cos θ}$ $f (θ) = 1 + \frac{2 [1 - \sqrt{2} \sin (θ + \frac{π}{4})]}{\sin 2 θ}$ $m a x . f (θ) = 1 + \frac{2 [1 - \sqrt{2}]}{1} = 3 - 2 \sqrt{2} \approx 0.17$ $a t θ = \frac{π}{4}$ $\Rightarrow m a x . A_{G r e e n} = (3 - 2 \sqrt{2}) a^{2} \approx 0.17 a^{2} \approx 0.34 r^{2}$
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