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Question
An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end.
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Solution
% In the figure, ELF is an arch whose width EF = 8 m and height = 2 m.
Let OX, OY be the coordinate axis. ELF is an ellipse with a = 4, b = 2
∴ Equation of ellipse,
`x^2/4^2 + y^2/2^2 = 1`
or `x^2/16 + y^2/4 = 1`
A point Q is at a distance of 1.5 m from end F.
∴ Distance of Q from O = 4 – 1.5 = 2.5 m
Let the height of the arch at point Q be P.
∴ P(2.5, P) lies on the ellipse.
∴ `(2.5)^2/16 + p^2/4 = 1`
or `p^2/4 = 1 - 6.25/16 = 9.75/16`
`p^2 = 9.75/16 xx 4 = 9.75/4`
`p = sqrt9.75/2`
= `3.122/2`
= 1.56 m (approximately)
Hence, the height of the arch at point Q = 1.56 m (approximately).
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