The figure below represents a design on the windows of a building. The curved part XY is an arc of a circle. The rise of the segmental arc is 10cm, its span is 100cm and XZ=ZY=120cm. calculate: (i) the radius of the circle (ii) the area of the segmental cap, correct to 2 significant figures. (iii) the total area of the design, correct to 3 significant figures.


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Question Number 204873 by necx122 last updated on 29/Feb/24

$T h e f i g u r e b e l o w r e p r e s e n t s a d e s i g n$ $o n t h e w i n d o w s o f a b u i l d i n g . T h e$ $c u r v e d p a r t X Y i s a n a r c o f a c i r c l e .$ $T h e r i s e o f t h e s e g m e n t a l a r c i s 10 c m,$ $i t s s p a n i s 100 c m a n d X Z = Z Y = 120 c m .$ $c a l c u l a t e :$ $(i) t h e r a d i u s o f t h e c i r c l e$ $(i i) t h e a r e a o f t h e s e g m e n t a l c a p,$ $c o r r e c t t o 2 s i g n i f i c a n t f i g u r e s .$ $(i i i) t h e t o t a l a r e a o f t h e d e s i g n, c o r r e c t$ $t o 3 s i g n i f i c a n t f i g u r e s .$
Commented by necx122 last updated on 29/Feb/24

Commented by necx122 last updated on 29/Feb/24
I will really need our help on this. Thanks in advance.
	Answered by A5T last updated on 29/Feb/24

	$C o n s t r u c t t h e p e r p e n d i c u l a r b i s e c t o r o f X Y,$ $t h e n Z a n d O, c e n t r e o f c i r c l e, l i e o n i t .$ $R^{2} = 50^{2} + {(R - 10)}^{2} \Rightarrow R^{2} = 50^{2} + R^{2} - 20 R + 100$ $\Rightarrow 20 R = 2600 \Rightarrow R = 130 c m$ $100^{2} = 2 \times 130^{2} - 2 \times 130^{2} c o s X O Y$ $\Rightarrow c o s X O Y = \frac{119}{169} \Rightarrow s i n X O Y = \frac{120}{169}$ $\Rightarrow A r e a o f s e c t o r = \frac{{s i n}^{- 1} (\frac{120}{169})}{360} \times 130^{2} π \approx 6671.9699$ $A r e a o f X O Y = 65 \times 130 \times \frac{120}{169} = 6000$ $\Rightarrow A r e a o f s e g m e n t a l a r c \approx 671.9699 {c m}^{2}$ $\Rightarrow T o t a l a r e a o f d e s i g n \approx 60 \times 120 \times \frac{5 \sqrt{119}}{72} - 671.9699$ $500 \sqrt{119} - 671.9699 \approx 4782.389$ $[100^{2} = 2 \times 120^{2} - 2 \times 120^{2} c o s X Z Y \Rightarrow c o s X Z Y = \frac{47}{72}$ $\Rightarrow s i n X Z Y = \frac{5 \sqrt{119}}{72}]$
	Commented by necx122 last updated on 29/Feb/24

	$T h i s i s c l e a r, u n d e r s t a n d a b l e a n d$ $p r e c i s e . T h a n k y o u s i r .$
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