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)

Made by

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.