Class 11

Math

3D Geometry

Conic Sections

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?

The equation of the parabola is of the form $x_{2}=4ay$ (as it is opening upwards)

Since, the parabola passes through point $(25 ,10)$

$(25 )_{2}=4a(10)$

$⇒a=4×4×1025 =325 $

So, the equation of arch of parabola is

$x_{2}=85 y$

When $y=2m$

$⇒x_{2}=45 $

$⇒x=45 m$

$∴AB=2×45 m=2×1.118m(approx)=2.23m(approx)$

So, the arch is $2$ m from the vertex of the parabola, its width is approximately $2.23$ m