Misc 4
An arch is in the form of semi ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 1.5 m from one end.
The arch is in form of semi−ellipse
It is 8m wide & 2m high
Let AB = width of semi-ellipse = 8 m
& CO = Height of semi-ellipse = 2 m
We know that
Length of Major axis of ellipse = 2a
AB = 2a
8 = 2a
2a = 8
a = 𝟖/𝟐 = 4
Length of semi-minor axis = b
CO = b
2 = b
b = 2
Now, equation of ellipse
𝑥2/𝑎2 + 𝑦2/𝑏2 = 1
Putting values of a & b
𝑥2/4^2 + 𝑦2/2^2 = 1
𝒙𝟐/𝟏𝟔 + 𝒚𝟐/𝟒 = 1
We need to find the height of arch at a point 1.5 m from one end.
Let y be the height of arch from a point 1.5 from one end.
Now, width of arch is (4 − 1.5m) = 2.5m
So, point P (2.5, y) lie on ellipse
it will satisfy equation of ellipse
Putting x = 2.5 & y = y in (1)
𝑥2/16 + 𝑦2/4 = 1
((2.5)2)/16 + 𝑦2/4 = 1
6.25/16 + 𝑦2/4 = 1
(6.25 + 4𝑦^2)/16 = 1
6.25 + 4y2 = 16
4y2 = 16 − 6.25
4y2 = 9.75
y2 = 9.75/4
y = √9.75/2
y = 3.12/2
y = 1.56 (approx)
Hence, the required height is 1.56 m (approx)

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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