An arc is in the form of a semi-ellipse. It is `8m` wide and `2m` high at the centre. Find the height of the arch at a point `1. 5c m` from one end.

Explanation

Let the equation of the ellipse of which the given arch is a part be
` x^(2)/ a^(2) + y^(2)/ b^(2) = 1.` …(i)
It is being given that
A'A = 8 m and OB = 2 m
` rArr 2a = 8 and b = 2`
` rArr a = 4 and b = 2.`
So, the equation of the arch is
` x^(2) / 16 + y^(2)/ 4 = 1.`
Let P be a point on A' OA such that AP = 1.5 m.

Draw `PM bot A'A`, meeting the ellipse at M.
Then, AP = 1.5 m (given).
Now, `OP = (OA - AP) = (4-1.5) m = 2.5 m. ` Let PM = p m.
So, the coordinates of M are (2.5, p).
Since M lies on the ellipse, we have
` ((2.5)^(2))/16 + p^(2)/ 4 = 1 rArr p^(2)/ 4 = (1-(625)/(1600)) rArr p^(2) = 39/16 `
` rArr p = sqrt(39)/4 = (6.24)/4 = 1.56`.
Hence, the height of the arch at a point 1.5 m from one end is ` 1.56` m.