Math Question and Answers


Question and Answers Forum

All Questions Topic List
Mensuration Questions
Previous in All Question Next in All Question
Previous in Mensuration Next in Mensuration
Question Number 123738 by ajfour last updated on 27/Nov/20

Commented by ajfour last updated on 27/Nov/20

$F i n d m i n i m u m o f \frac{R}{s} .$ $s i s t h e s i d e o f t h e p i n k s h a d e d$ $e q u i l a t e r a l t r i a n g l e .$
	Answered by mr W last updated on 28/Nov/20

	Commented by mr W last updated on 28/Nov/20

	$A D = \frac{2 R \sin α}{\sin (α + β)}$ $∠ A B D = α + \frac{π}{3}$ $A B = \frac{2 R \sin α \sin (α + β + \frac{π}{3})}{\sin (α + β) \times \sin (α + \frac{π}{3})}$ $s = \frac{2 R \sin α \sin β}{\sin (α + β) \times \sin (α + \frac{π}{3})}$ $\Rightarrow λ = \frac{s}{2 R} = \frac{\sin α \sin β}{\sin (α + β) \times \sin (α + \frac{π}{3})} . . . (I I)$ $A E = 2 R \cos β$ $E D = 2 R [\cos β - \frac{\sin α}{\sin (α + β)}]$ $E B = \frac{2 R}{\sin \frac{π}{3}} [\cos β - \frac{\sin α}{\sin (α + β)}] \sin (α + β + \frac{π}{3})$ $\Rightarrow \frac{E B}{2 R} = \frac{2}{\sqrt{3}} [\cos β - \frac{\sin α}{\sin (α + β)}] \sin (α + β + \frac{π}{3})$ ${E B}^{2} = {[\frac{2 R \sin α \sin (α + β + \frac{π}{3})}{\sin (α + β) \times \sin (α + \frac{π}{3})}]}^{2} + {[2 R \cos β]}^{2} - 2 \times \frac{2 R \sin α \sin (α + β + \frac{π}{3})}{\sin (α + β) \times \sin (α + \frac{π}{3})} \times 2 R \cos β \times \cos β$ ${(\frac{E B}{2 R})}^{2} = {[\frac{\sin α \sin (α + β + \frac{π}{3})}{\sin (α + β) \times \sin (α + \frac{π}{3})}]}^{2} + \cos^{2} β - \frac{2 \sin α \cos^{2} β \sin (α + β + \frac{π}{3})}{\sin (α + β) \times \sin (α + \frac{π}{3})}$ $\Rightarrow {[\frac{\sin α \sin (α + β + \frac{π}{3})}{\sin (α + \frac{π}{3}) \sin (α + β)}]}^{2} + \cos^{2} β - \frac{2 \sin α \cos^{2} β \sin (α + β + \frac{π}{3})}{\sin (α + \frac{π}{3}) \sin (α + β)}$ $= \frac{4}{3} {[\cos β \sin (α + β) - \sin α]}^{2} {[\frac{\sin (α + β + \frac{π}{3})}{\sin (α + β)}]}^{2} . . . (I)$ $w e g e t f r o m (I I) a n d (I) g r a p h i c a l l y$ $λ_{m a x} = \frac{s_{m a x}}{2 R} \approx 0.1831$ $a t α \approx 16.5 ° a n d β \approx 24 °$
	Commented by mr W last updated on 28/Nov/20

	Commented by ajfour last updated on 30/Nov/20

	$T h a n k s f o r s o l v i n g!$ $M a r v e l l o u s l y d o n e .$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum