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Misc 3 - The cable of a uniformly loaded suspension bridge hangs - Parabola - Beam problem

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  1. Chapter 11 Class 11 Conic Sections
  2. Serial order wise
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Misc 3 The cable of a uniformly loaded suspension bridge hangs in the form of a parabola. The roadway which is horizontal and 100 m long is supported by vertical wires attached to the cable, the longest wire being 30 m and the shortest being 6 m. Find the length of a supporting wire attached to the roadway 18 m from the middle. Cable is always facing upwards with the axis vertical Since, the axis is positive y-axis, its equation is x2 = 4ay First, we find coordinates of point B Given length of cable = 100 m Hence, AB = 100 m So, BC = ﷐𝐴𝐵﷮2﷯ = ﷐100﷮2﷯ = 50 m Also, DB = 30 – 6 = 24 m Hence point B is B (50, 24) Now, Since point B(50, 24) lies on the parabola Putting x = 50, y = 24 in equation of parabola x2 = 4ay (50)2 = 4a (24) 2500 = 4a × 24 4a × 24 = 2500 4a = ﷐2500﷮24﷯ 4a = ﷐625﷮6﷯ a = ﷐𝟔𝟐𝟓﷮𝟐𝟒﷯ Now, we need to length of a supporting wire attached to the roadway 18 m from the middle Hence OR = 18 m, We need to find QP Let QP = d So, QR = QP – 6 = d – 6 So, coordinates of point Q is Q(18, d – 6) So, coordinates of point Q is Q(18, d – 6) Since point Q lies on parabola it will satisfy the equation of parabola Equation of parabola is x2 = 4ay Putting x = 18 & y = d – 6 & a = ﷐625﷮24﷯ 182 = 4 ﷐﷐625﷮24﷯﷯ (d – 6) 18 × 18 = 4 ﷐﷐625﷮24﷯﷯ (d – 6) 18 × 18 × ﷐24﷮4 × 625﷯ = (d – 6) (d – 6) = 18 × 18 × ﷐24﷮4 × 625﷯ (d – 6) = 3.11 d = 6 + 3.11 d = 9.11 So, QP = 9.11 m ∴ Length of support is 9.11 m.

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