Given an arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre.

Equation of the ellipse is x² /a² + y² /b² = 1 ...........1   (where a > b)

Now from figure,

OA = a = 4

and OB = b = 2

From equation 1, we get

      x² /4² + y² /2² = 1

=> x² /16 + y² /4 = 1

Again from figure,

CA = 1.5

Then OC = 4 - 1.5 = 2.5

Let CP = h

So coordinate of point P is (2.5, h)

Since this point lies on the semi-ellipse,

(2.5)² /16 + h² /4 = 1

=> 6.25/16 + h² /4 = 1

=> h² /4 = 1 - 6.25/16

=> h² /4 = (16 - 6.25)/16

=> h² /4 = (9.75)/16

=> h²  = (9.75)/4

=> h²  = 2.44

=> h = √2.44

=> h = 1.562  (approximately)

So the height of the arch at a point 1.5 m from one end is h = 1.562 m  (approximately)