Question

In a model, it is shown that an arc of a bridge is semi-elliptical with major axis horizontal. if the length of the base is 9m and the highest part of the bridge is 3m from the horizontal; the best approximation of the height of the arch at 2m from the centre of the base is

Answer 1

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Answer 2

Concept: The locus of all those locations in a plane whose sum of distances from two fixed points in the plane is constant is called an ellipse. The foci (singular focus) are the fixed points that are encircled by the curve. The constant ratio is the eccentricity of the ellipse, and the fixed line is the directrix. Eccentricity is a property of the ellipse that indicates its elongation and is symbolized by the letter 'e.'

Given: Length of the major axis = 9m

           Length of the semi minor axis = 3m

           x = 2m

To find: Best approximation of the height of the arch from the centre of the base

Solution:

Let the equation of the semi elliptical arc be $\frac{x^{2} }{a^{2} } + \frac{y^{2} }{b^2} = 1 (y > 0)$

Length of the major axis = 2a = 9 => a= 9/2

Length of the semi minor axis = b = 3m

Therefore, the equation of the arc becomes,

$\frac{4x^{2} }{81} + \frac{y^{2} }{9} = 1$

If x= 2, then

$\frac{4(2)^2}{81} +\frac{y^2}{9} =1\\\frac{16}{81} +\frac{y^2}{9} =1\\\frac{y^2}{9} = 1- \frac{16}{81}\\\frac{y^2}{9} =\frac{65}{81} \\y^2 = \frac{65}{9} \\y = \sqrt{\frac{65}{9} } \\y = \frac{1}{3} \sqrt{65} \\y = \frac{8}{3} approximately$

Hence, the best approximation of the height of the arch at 2m from the centre of the base is $\frac{8}{3} m$.

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