Q. 13

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

Answer :

Given: Top of the towers are 30 m above the roadway and are 200 m apart. Cable is 5 m above the roadway at center.

Need to find: Length of the vertical supporting cable, 30 m from the center.

A and B are the top of the towers. AE and BF are the height of the towers. H is the center of the bridge. HI is the 5 m above from the roadway.

Let, the equation of the parabola be: x^{2} = 4a(y – b)

Here b = 5. So, x^{2} = 4a(y – 5)

Here, AB = 200 m and BF = 30 m.

So, the coordinate of the point B is (100, 30)

The point is on the parabola.

Hence, x^{2} = 4a(y – 5)

⇒ 10000 = 4a (30 – 5)

⇒ 10000 = 4a x 25

⇒ a = 100

Now we need to find, the length of the vertical supporting cable, 30 m from the center.

The x-coordinate of the point, 30 m from the center, is 30.

So, 30 x 30 = 4a (y – 5)

⇒ 900 = 400 (y – 5)

⇒ y – 5 =

⇒ y =

So, the length of the vertical supporting cable is m = 7.25 m

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RS Aggarwal - Mathematics