Question

A cable suspended between two poles which are 200 ft apart has a sag of 50 ft. If the cable hangs in a form of a parabola, find its equation with the origin at its lowest point.

Answer 1

Answer:

Let the equation of the parabola be x2=4ay by taking lowest point as the origin & vertical line through it as y-axis.

Now, the points of both the supports will be at (−300,100) & (300,100) which lie on the parabola hence, setting x=300 & y=100 in the equation we get

(300)2=4a(100)

4a=90000100=900

Hence, setting the value of 4a, the equation of the parabola is

x2=900y

Answer 2

Answer:

The equation of parabola to the lowest point with its origin is $x^{2} = 200y$.

Concept:

An equal distance from the focus, a fixed point, and the directrix, a fixed line, is present at every point on a parabola. The intersection of a plane parallel to a cone element and a right circular cone results in a plane curve when a point moves in such a way that the distance it travels from a fixed point equals the distance it travels from a fixed line.

Given:

Suspension between two poles = 200 ft

Cable has a sag = 50 ft

Find:

What is the lowest point with the origin?

Solution:

As we know that equation of parabola, $x^{2} =4ay$ which is lowest point as the origin and vertical line is passing through y-axis.

Now, the points which lie on parabola are (-100, 50) and (100, 50).

In this case, we must use x = 100 and y = 50.

We will substitute this values in $x^{2} =4ay$, we get:

$100^{2} = 4a(50)\\4a = \frac{10000}{50} \\4a = 200$

So, 4a = 200.

Hence, the equation of parabola is $x^{2} = 200y$.

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