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

The given arch is in the form of semi-ellipse which is 8 m wide and 2 m high at the center.

So, the length of major axis is 8 m and the length of semi minor axis is 2 m.

Take origin of the coordinate plane at the center of ellipse and major axis along x axis.

The equation of the semi ellipse is,

x 2 a 2 + y 2 b 2 =1(1)

Since length of major axis is 2a,

2a=8 a=4

Since length of semi minor axis is b,

b=2

Substitute values of a and b in (1),

x 2 4 2 + y 2 2 2 =1(2)

Let A be the point on major axis such that AB=1.5 m. Draw AC⊥OB.

Now, length of OA is,

OA=( 4−1.5 ) =2.5

Therefore, xcoordinate is 2.5.

Substitute x coordinate in (2) to determine ycoordinate,

( 2.5 ) 2 16 + y 2 4 =1 y 2 4 =1− ( 2.5 ) 2 16 y 2 =4( 9.75 16 ) y=1.56

Thus, the height of the arch at the required point is 1.56 m.