Q.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.

Concept: Equation of an ellipse and coordinate geometry.

Place the origin at the center of the base of the arch. The semi-ellipse has its major axis along the $x$-axis (width) and minor axis along the $y$-axis (height).

The arch is $8$ m wide, so the semi-major axis $a = 4$ m. It is $2$ m high at the center, so the semi-minor axis $b = 2$ m. The equation of the ellipse is

A point $1.5$ m from one end means $1.5$ m from $x = -4$, which places it at $x = -4 + 1.5 = -2.5$ m.

Substitute $x = -2.5$ into the ellipse equation:

$\frac{(-2.5)^2}{16} + \frac{y^2}{4} = 1 \implies \frac{6.25}{16} + \frac{y^2}{4} = 1$ …

The problem asks us to find a specific height on a semi-elliptical arch. The natural approach is to set up a coordinate system, write the equation of the ellipse, and then use the given horizontal distance to compute the corresponding vertical coordinate.

An ellipse centred at the origin with horizontal semi-axis $a$ and vertical semi-axis $b$ has equation

Since the arch is a semi-ellipse (the upper half), we take $y \geq 0$.

Setting up the coordinate system

Place the origin at the centre of the base of the arch. The arch is $8$ m wide, so it extends from $x = -4$ m to $x = 4$ m. The height at the centre (where $x = 0$) is $2$ m, which is the maximum height of the semi-ellipse.

From this geometry:

• The semi-major axis (horizontal) is $a = 4$ m.
• The semi-minor axis (vertical) is $b = 2$ m.

The equation of the ellipse becomes

Finding the point of interest

The phrase "a point $1.5$ m from one end" means $1.5$ m horizontally from one of the two ends of the arch. Take the left end at $x = -4$ m. Moving $1.5$ m to the right gives

(Alternatively, measuring from the right end $x = 4$ m would give $x = 4 - 1.5 = 2.5$ m, which by symmetry yields the same height.)

Computing the height

Substitute $x = -2.5$ into the ellipse equation: …