Answer :

Given that the cable hangs in the form of a parabola.

It is told that the length of the shortest wire supported is 6.

It is clear that the vertex of the parabola is S(0, 6).

We know that the equation of the parabola having (0, a) as vertex is x^{2} = 4b(y - a)

Let us assume the equation of the parabola is x^{2} = 4b(y - 6).

It is told that the road way is 100m long. We usually give maximum support at the middle of the road i.e. at 50m. At the 50m of the road way, the length of the support wire used is 30m.

It is clear that the point (50, 30) lies on parabola, substituting this point in the equation of the parabola, we get,

⇒ (50)^{2} = 4b(30 - 6)

⇒ 2500 = 4b(24)

⇒ 96b = 2500

The equation of the parabola is .

We need to find the length of the support that is needed to give at the 18m in the roadway.

Let us assume the length of the support is l m,

We have a point (18, l) on the parabola, substituting in the equation we get,

⇒

⇒

⇒ 3.11 = l - 6

⇒ l = 9.11m

∴The length of the support required is 9.11m.

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