Question

A grider of a railway bridge is a parabola with its vertex at the heighest point Iom above the ends . If the span is 100m , find its height 20m from the midpoint?

Answer 1

The height from the midpoint is 8.4 m

Given:

Vertex of the parabola = 10 m

Span of the parabola = 100 m

To find:

Height 20 m from the midpoint = ?

Solution:

The diagram of the parabola with points are attached in the image below.

The parabola is in y axis.

So, the equation of the parabola is:

$\bold{x^2=-4ay}$

Let 'O' be the vertex of the parabola.

From question, OM = 10 m and PQ = 100 m

Let 'M' be the midpoint of PQ.

The distance of 'MQ' is 50 m. The co-ordinate of the point Q is (50, -10).

Now, the equation of the parabola is:

$\bold{(50)^2=-4a(-10)}$

$\bold{2500=40a}$

$\bold{\therefore a=62.5}$

Now, the height of a point 20 m from midpoint:

Let the point be 'A' and the coordinate of the point be $\bold{(20,-y1)}$.

Now, the equation of the parabola becomes,

$\bold{(20)^2=-4\times62.5\times (-y1)}$

$\bold{y1=\frac{400}{250}}$

$\bold{\therefore y1=1.6}$

Now, the required height is:

$\bold{RA=RS-AS= 10-1.6}$

$\bold{\therefore RA=8.4 \ m}$

Answer 2

Step-by-step explanation:

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