Question

An arch is in the form of a semi-ellipse. It is m wide and m high at the centre. Find the height of the arch at a point m from one end.

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Solution

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Since the height and width of the arc from the centre is m and m respectively. It is clear that the length of the major axis is m, while the length of the semi-minor axis is  m

The origin of the coordinate plane is taken as the centre of the ellipse while the major axis is taken along the -axis.  Hence the semi-ellipse can be diagrammatically represented as,

The equation of the semi-ellipse will be of the form  where is the semi-major axis.

Accordingly,  



Therefore, the equation of the semi-ellipse is   ...(1)

Let be a point on the major axis such that m

Draw



The -coordinate of point is .

On substituting the value of with in equation (1), we obtain











(approx)  



Thus the height of the arch at a point m from one end is approximately .

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