Misc 2
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?
Arch is downwards,
Since, the axis of parabola is negative y-axis,
its equation is
x2 = 4ay
First, we find coordinates of point B
Given,
Width of Parabola = AB = 5m
So, BC = 5 2 m
Also, Parabola is 10 m high
Hence,
OC = 10 m
BD = OC = 10m
Hence, coordinates of point B is ,
Now since point B 5 2 , 10 lies on the parabola
It will satisfy the equation of the parabola,
Putting x = 5 2 , y = 10 in equation
x2 = 4ay
5 2 2 = 4a ( 10)
25 4 1 4 1 10 = a
25 4 1 4 1 10 = a
5 4 4 2 = a
5 32 = a
a =
Hence, the equation of parabola is
x2 = 4ay
Putting a = 5 32
x2 = 4 5 32 y
x2 =
We need to find how wide the arch is it 2 m from the vertex of the parabola
Let point P be point 2 m
from vertex of parabola
Hence,
OP = 2m
We need to find MN
Now equation of parabola is
x2 = 5 8
Putting y = 2
x2 = 4 5 32 ( 2)
x2 = 4 5 2 32
x2 = 5 4
x = 5 4
x =
Hence,
PN = 5 2
So, MN = 2PN
= 2 5 2 = 5
Thus, the width of the arch MN = m
= 2.23 m (approx.)

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.