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Q.A cable of a suspension bridge is in the form of a parabola whose span is 40 mts. The road way is 5 mts below the lowest point of the cable. An extra support is provided across the cable 30 mts above the ground level. Find the length of the support if the height of the pillars are 55 mts.

Tamil Nadu DgeTamil Nadu HSC (DGE) Board 2018Subjective· 10mImportance★★★★★
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Model the cable as a parabola with vertex at its lowest point; use the pillar-top coordinates to find the parabola's constant, then find the half-width at the support's height and double it.

  1. Take the origin at the vertex VV (the lowest point of the cable), with the XX-axis horizontal and YY-axis vertical, parabola opening upward: X2=4aYX^2=4aY.
  2. Span of the bridge =40=40 m, so by symmetry the two pillars are at X=−20X=-20 and X=20X=20.
  3. The roadway is 55 m below the vertex of the cable, so — taking the roadway (ground) as the reference level — the vertex sits at height 55 m above the road.
  4. The pillars are 5555 m tall (measured from road level), and the cable is attached at the pillar tops, so the cable's height there is 5555 m above the road, i.e. 55−5=5055-5=50 m above the vertex VV.
  5. So the point X=20X=20 on the parabola has Y=50Y=50 (height measured from the vertex). Substitute into X2=4aYX^2=4aY: 202=4a(50)⇒400=200a⇒a=220^2=4a(50)\Rightarrow400=200a\Rightarrow a=2.
  6. The parabola's equation (relative to the vertex) is X2=8YX^2=8Y.
  7. The extra support is at 3030 m above ground level, i.e. 30−5=2530-5=25 m above the vertex: Y=25Y=25. …
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