Question

An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the center. 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 center 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 \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, y ≥ 0\) , where a is the semi-major axis 
Accordingly, \(2a = 8 ⇒ a = 4 b = 2 \)
Therefore, the equation of the semi-ellipse is \(\frac{x^2}{16} +\frac{ y^2}{4} = 1, y ≥ 0 ......(1)\)

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
\(\frac{(2.5)^2}{16} +\frac{ y^2}{4} = 1\)

\(\frac{6.25}{16} + \frac{y^2}{4} = 1\)

\(y^2 = 4 (1 – \frac{6.25}{16})\)

\(= 4 (\frac{9.75}{16})\)

\(= 2.4375\)
\(y = 1.56 (approx.)\)

\(∴AC = 1.56 m \)

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