1. The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which si horizontal & 100m long is supported by vertical wires attached to the cable , the longest wire being 30m and shortest being 6m.
Finnd the length of a supporting wire attatched to the roadway 18m from the middle?

Asked by gargpurvi85 | 28th Feb, 2015, 08:13: PM

Expert Answer:

C o n s i d e r space t h e space f o l l o w i n g space f i g u r e.
 
G i v e n space t h a t space t h e space c a b l e space h a n g s space i n space t h e space f o r m space o f space a space p a r a b o l a. L e t space C A B space b e space t h e space c a b l e space a n d space X apostrophe O X space b e space t h e space r o a d w a y. C l e a r l y comma space t h e space c a b l e space i s space i n space t h e space s h a p e space o f space a space p a r a b o l a space w i t h space v e r t e x space a t space. L e t space O space b e space t h e space c e n t r e space o f space t h e space r o a d w a y. S i n c e space t h e space c a b l e space i s space s u p p o r t e d space b y space v e r t i c a l space w i r e space o f space 6 space m comma space t h e space c o o r d i n a t e s space o f space A space a r e comma space A open parentheses 0 comma 6 close parentheses. T h u s comma space t h e space e q u a t i o n space o f space t h e space p a r a b o l a space i s x squared equals 4 a open parentheses y minus 6 close parentheses A l s o space g i v e n space t h a t space t h e space l e n g t h space o f space t h e space l o n g e s t space v e r t i c a l space w i r e space s u p p o r t i n g space t h e space c a b l e space i s space 30 space m. S i n c e space t h e space l e n g t h space o f space t h e space r o a d y w a y space i s space 100 space m comma space w e space h a v e comma M N equals 100 space m space rightwards double arrow O M equals 50 space m H e n c e space t h e space c o o r d i n a t e s space o f space B space a r e comma space B left parenthesis 50 comma 30 right parenthesis space a n d space t h e space c o o r d i n a t e s space o f space C space a r e comma space C left parenthesis negative 50 comma 30 right parenthesis T h e space p a r a b o l a space p a s s e s space t h r o u g h space B space a n d space C rightwards double arrow x squared equals 4 a open parentheses y minus 6 close parentheses rightwards double arrow 50 squared equals 4 a open parentheses 30 minus 6 close parentheses rightwards double arrow 2500 equals 4 cross times a cross times 24 rightwards double arrow a equals fraction numerator 2500 over denominator 4 cross times 24 end fraction W e space n e e d space t o space f i n d space t h e space l e n g t h space o f space s u p p o r t i n g space w i r e space a t space t h e space p o i n t space 18 space m space f r o m space t h e space c e n t r e. T h a t space i s space w e space n e e d space t o space f i n d space t h e space y space c o o r d i n a t e space o f space t h e space p o i n t space w h o s e space x space c o o r d i n a t e space i s space 18. L e t space P space b e space P open parentheses 18 comma y close parentheses space S i n c e space i t space l i e s space o n space t h e space p a r a b o l a comma space w e space h a v e comma x squared equals 4 a open parentheses y minus 6 close parentheses rightwards double arrow 18 squared equals 4 a open parentheses y minus 6 close parentheses rightwards double arrow 324 equals 4 cross times fraction numerator 2500 over denominator 4 cross times 24 end fraction open parentheses y minus 6 close parentheses rightwards double arrow fraction numerator 324 cross times 24 over denominator 2500 end fraction equals y minus 6 rightwards double arrow y minus 6 equals 3.1104 rightwards double arrow y equals 6 plus 3.1104 rightwards double arrow y equals 9.1104 space m H e n c e space t h e space l e n g t h space o f space t h e space v e r t i c a l space s u p p o r t i n g space w i r e comma space 18 space m space a w a y space f r o m space t h e space c e n t r e o f space t h e space b r i d g e space i s space 9.1104 space m

Answered by Vimala Ramamurthy | 1st Mar, 2015, 01:19: PM

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