Question

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

• A. $3$
• B. $4$
• C. $\sqrt{5}$
• D. $\frac{9}{3}$

Answer 1

The correct option is C $\sqrt{5}$

Let the vertex of the parabola be at the origin and axis be along OY. Then, the equation of the parabola is :

$x^2=4ay$

The coordinates of end A of the arc are (2.5, 10) and it lies on the parabola.

Therefore,

$(2.5{)}^2=4a\times 10$

$a=\frac{6.25}{40}=\frac{625}{4000}=\frac{5}{32}$

Thus, the equation of the parabolic arc is $x^2=\frac{5}{8}y$.

When $y=2$,

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

$x=\frac{\sqrt{5}}{2}m$

Hence, the width of the arc at a height of 2 m from the vertex is $2\times \frac{\sqrt{5}}{2}m=\sqrt{5}m$