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.

Answer 1

% 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).