Question

The tower of a bridge, hung in the form of a parabola have their tops 30 meters above the roadway and are 200 meters apart. If the cable is 5 meters above the roadway at the centre of the bridge, find the length of the vertical supporting cable 30 meters from the centre.

Answer 1

Let CAB be the cable of the bridge and X'OX be the roadway.

Let A be the centre of the bridge.

From the figure, vertex of parabola is at A(0, 5).

Let the equation of parabola be

x² = 4b (y – 5)  ...(i)

Since the parabola passes through (100, 30).

Substituting x = 100 and y = 30 in (i), we get

100² = 4b (30 – 5)

∴ 100² = 4b(25)

∴ 100² = 100b

∴ b = `(100 xx 100)/100`

∴ b = 100

Substituting the value of b in (i), we get

x² = 400(y – 5) ...(iii)

Let l metres be the length of vertical supporting cable.

Then P(30, l) lies on (ii).

∴ 30² = 400 (l – 5)

∴ 900 = 400 (l – 5)

∴ `9/4` = l – 5

∴ l = `9/4 + 5`

∴ l = `29/4"m"`

∴ l = 7.25 m

∴ The length of vertical supporting cable is 7.25 m.