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$A space p i e c e space o f space w i r e space 10 c m space l o n g space i s space f o r m e d space i n t o space t h e space s h a p e space o f space a space s e c t o r space o f space a space c i r c l e space o f space r a d i u s space r space c m space a n d space a n g l e space theta space r a d i a n s : left parenthesis i right parenthesis space s h o w space t h a t : space theta equals fraction numerator 10 minus 2 r over denominator r end fraction left parenthesis i i right parenthesis space s h o w space t h a t space a r e a space o f space s e c t o r space A equals 5 r minus r squared$

Asked by Olivia Adhikary | 4th Aug, 2014, 12:53: PM

Expert Answer:

$left parenthesis i right parenthesis space S i n c e space t h e space p i e c e space o f space w i r e space 10 space c m space l o n g space i s space f o r m e d space i n t o space t h e space s h a p e r space o f space a space s e c t o r comma space t h e space p e r i m e t e r space o f space t h e s e c t o r space equals 10 space c m C o n s i d e r space t h e space f o l l o w i n g space d i a g r a m.$

$F r o m space t h e space d i a g r a m comma space p e r i m e t e r space o f space t h e space a r c equals O A plus O B plus A B equals l L e t space apostrophe r apostrophe space b e space t h e space r a d i u s space o f space t h e space s e c t o r space a n d space l e t space A B space b e space t h e space l e n g t h space o f space t h e space a r c space A B. T h u s comma space r plus r plus l equals 10 rightwards double arrow l equals 10 minus 2 r A r c space l e n g t h space c a n space a l s o space b e space s t a t e d space a s space theta cross times r rightwards double arrow theta cross times r equals 10 minus 2 r rightwards double arrow theta equals fraction numerator 10 minus 2 r over denominator r end fraction$

$open parentheses i i close parentheses space A r e a space o f space t h e space s e c t o r equals fraction numerator theta r squared over denominator 2 end fraction rightwards double arrow A r e a space o f space t h e space s e c t o r equals fraction numerator begin display style fraction numerator 10 minus 2 r over denominator r end fraction end style r squared over denominator 2 end fraction...... open square brackets f r o m space p r e v i o u s space s u b p a r t comma space theta equals fraction numerator 10 minus 2 r over denominator r end fraction close square brackets rightwards double arrow A r e a space o f space t h e space s e c t o r equals open parentheses fraction numerator 10 minus 2 r over denominator 2 end fraction close parentheses r rightwards double arrow A r e a space o f space t h e space s e c t o r equals 5 r minus r squared$

Answered by Vimala Ramamurthy | 5th Aug, 2014, 10:29: AM

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