An arch is in the form of a semi-ellipse. It is 8 m wide and 2 m high at the centre. Find the height of the arch at a point 2.5 m from one end.

Asked by Topperlearning User | 13th Nov, 2016, 07:23: PM

Expert Answer:

The arch is in the form of a semi-ellipse. Hence let AA' be the arch, and

 



AA' = 8 m. or 2a = 8 a = 4 m.

The arch is 2m heigh hence OB = 2m. We have to find the height of the arch from one end at a distance of 2.5 m. Let AL be 2.5 m. We have to find the height LM.

OA = 4 m, AL = 2.5 m OL = 1.5 m.

The equation of the ellipse is

 

x squared over a squared plus y squared over b squared equals 1

L e t space t h e space c o minus o r d i n a t e s space o f space M space a r e space left parenthesis 1.5 comma space h right parenthesis. space S u b s t i t u t i n g space t h e space v a l u e s comma space w e space g e t

fraction numerator 9 over denominator 4 cross times 16 end fraction plus h squared over 4 equals 1

rightwards double arrow h squared over 4 equals 1 minus 9 over 64 equals 55 over 64
rightwards double arrow h equals square root of 55 over 16 end root
H e n c e comma space t h e space h e i g h t space o f space t h e space a r c space a t space a space d i s tan c e space space 2.5 space c m space f r o m space o n e space e n d space i s space square root of 55 over 16 end root space m.

Answered by  | 13th Nov, 2016, 09:23: PM