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.

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 represented as:

(x² / a²) + (y² / b²) = 1, y ≥ 0

where a is the semi-major axis.

Accordingly, 2a = 8 ⇒ a = 4 and 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) = 2.5 m.

The x-coordinate of point C is 2.5.

On substituting x = 2.5 in equation (1), we get:

(2.5)² / 16 + y² / 4 = 1

⇒ 6.25 / 16 + y² / 4 = 1

⇒ y² / 4 = 1 − 6.25 / 16

⇒ y² / 4 = 9.75 / 16

⇒ y² = 2.4375

⇒ y = 1.56 (approx.)

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