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

Its form is of the shape of a parabola.

Let OX, OY be its coordinate axis, and the equation is y² = 4ax.

Height of arch, OL = 10 m

Width EF = 5 m

LF = `1/2` EF = `1/2 xx 5 = 5/2`

Coordinates of point F `(10, 5/2)`

Since the point `(10, 5/2)` lies on the parabola y² = 4ax

∴ `(5/2)^2 = 4a xx 10` or `40a = 25/4`

∴ 4a = `25/4 xx 1/10 = 5/8`

∴ Equation of parabola y² = `5/8 x`

2 m below top O, let the width of the arch be 2b.

∴ PM = `1/2 "PQ" = 1/2 xx 2"b" = "b"`

P has coordinates of the point (2, b) which lies on the parabola `"y"^2 = 5/8 "x"`.

∴ `"b"^2 = 5/8 xx 2 = 5/4`

∴ b = `sqrt5/2`

The width of the arch at this location,

= `2"b"`

= `2 xx sqrt5/2`

= `sqrt5` meter

= 2.24 meters (approximately)