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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?
Solution
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)
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