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प्रश्न
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.
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उत्तर
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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