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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.
Solution
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.
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