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 form 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 positive $y$-axis.
The equation of the parabola is of the form $x^2=4ay$ (as it is opening upwards)
Since, the parabola passes through point $(\frac{5}{2},10)$
${\left(\frac{5}{2}\right)}^2=4a\left(10\right)$
$\Rightarrow a=\frac{25}{4\times 4\times 10}=\frac{5}{32}$
So, the equation of arch of parabola is

$x^2=\frac{5}{8}y$

When $y=2m$
$\Rightarrow x^2=\frac{5}{4}$
$\Rightarrow x=\sqrt{\frac{5}{4}}m$
$\therefore AB=2\times \sqrt{\frac{5}{4}}m=2\times 1.118m\left(approx\right)=2.23m\left(approx\right)$
So, the arch is $2$ m from the vertex of the parabola, its width is approximately $2.23$ m