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.)

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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