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

Since the height and width of the arc from the centre is 2 m and 8 m respectively, it is clear that the length of the major axis is 8 m, while the length of the semi-minor axis is 2 m. The origin of the coordinate plane is taken as the centre of the ellipse, while the major axis is taken along the x-axis. Hence, the semi-ellipse can be diagrammatically represented as

The equation of the semi-ellipse will be of the form X²/a² + Y²/b² =1.y>()  where a is the semi-major axis
Accordingly, 2a = 8 ⇒ a = 4 b = 2

Therefore, the equation of the semi-ellipse is X2/16 + Y2/4 =1.y> 0            ....(1)
Let A be a point on the major axis such that AB = 1.5 m.
Draw AC⊥ OB.
OA = (4 – 1.5) m = 2.5 m
The x-coordinate of point C is 2.5.
On substituting the value of x with 2.5 in equation (1), we obtain

Thus, the height of the arch at a point 1.5 m from one end is approximately 1.56 m.