Question

An arch is in the form of a parabola with its axis vertical. The arch is 10 m high and 5 m wide at the base. How wide is it 2 m from the vertex of the parabola?

Answer 1

The origin of the coordinate plane is taken at the vertex of the arch in such a way that its vertical axis is along the positively-axis.
This can be diagrammatically represented as

The equation of the parabola is of the form $x^ 2 = 4ay$ (as it is opening upwards).
It can be clearly seen that the parabola passes through point $(\frac{5}{2}, 10)$

$(\frac{5}{2})^2 = 4a(10)$

$4a = \frac{25}{(4\times4\times10)}$

$⇒ a =\frac{ 5}{32}$

Therefore, the arch is in the form of a parabola whose equation is $x^2 = \frac{5}{8} (2)$

When $y = 2 m, x^2 = \frac{5}{8} \times(2)$

$⇒ x^2 = \frac{5}{4}$

$x = \sqrt{(\frac{5}{4})} = \sqrt{\frac{5}{2}}$

$∴ AB = 2 \times \sqrt{\frac{5}{2}}m = \sqrt{5}m = 2.23m$(approx.)

Hence, when the arch is 2 m from the vertex of the parabola, its width is approximately 2.23 m.