 Hello and welcome to the session let us discuss the following question it says the cable of a uniformly loaded suspension which hangs in the form of a parabola. The roadway which is horizontal and 100 meter long is supported by vertical wires attached to the cable, the longest wire being 30 meter and the shortest being 6 meter. In the length of a supporting wire attached to the roadway 18 meter from the middle. So here we are given a wire hanged in the form of a parabola and the horizontal roadway which is 100 meter long is supported by vertical wires the longest wire has length 30 meter and the shortest has length 6 meter and we have to find the length of the supporting wire attached to the roadway 18 meter from the middle. Let us now move on to the solution. Now here we can see that the wire which is hanged in the shape of a parabola has axis of symmetry as y axis so the equation of the parabola is given by x square is equal to 4 a y the length of the longest wire o f is 30 meter and the length of the shortest wire a a dash or b b dash is equal to 6 meter which is also equal to f f dash. Now the length of the roadway which is a b which is equal to a dash b dash is given to be 100 meter so here the coordinate of the point a dash is minus x y and the coordinate of the point b dash will be x y and that means the length of the roadway a dash b dash or a b is 2 x and this implies here x is equal to 50 so here x has value 50. Now when x is equal to 50 y is 30 minus 6 meter this point this whole minus this that is 24 meter so the coordinate of y in this condition is equal to o f minus f f dash that is 30 minus 6 that is 24 so here the coordinate of b dash are 50 24 thus we substitute the value of x and y in the equation of the parabola so 50 square is equal to 4 a y where y is 24 and this implies a is equal to 2500 upon 4 into 24 which is equal to 625 upon 24. Now we have to find the length of the supporting wire attached 18 meter from the middle of the roadway that means we have to find the length c d let this point be c dash now c dash f dash is given to be 18 meter that means x is 18 at this point right now c d length of c d is equal to c c dash plus c dash d c c dash is 6 meter so it is 6 plus c dash d meter now length of c dash d is same as the coordinate of y where x is equal to 18 because your x is 18 so the length of c d which is c c dash plus c dash d will be the coordinate of y right now your x is 18 a is equal to 625 upon 24 so we substitute this in the equation of the parabola to obtain the value of y which is the length of c dash d so 18 square is equal to 4 a which is 625 upon 24 into y and this implies 24 is equal to 625 upon 6 into y and this implies y is equal to 324 into 6 upon 625 which is equal to 3.1104 now length of c d is equal to 6 plus c dash d which is y so it is 3.104 meter which is equal to 9.11 meter approximately hence the length of the wire is 9.11 meter approximately and this completes the question bye for now take care have a good day