the arch of a bridge has the shape of half an ellipse. if the span of the arch is 40m and its height at the highest point is 15m, find the height of the arch at a point 10m from the center.
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Take the mid-point of the base as the centre C(0,0).
Since the base is 40ft, the vertices A and A
′
are (20,0) and (−20,0).
Therefore 2a=40
⇒ a=20,b=16
Equation of the ellipse is
a
2
x
2
+
b
2
y
2
=1
∴
400
x
2
+
256
y
2
=1
Let y
1
be the height of the arch from 9ft right of the centre.
∴ (9,4) is a point on the semi-ellipse.
The equation of ellipse is
a
2
+b
2
x
2
+y
2
=1
⇒
400
9
2
+
256
y
1
2
=1
⇒
400
9
2
+
256
y
1
2
=1
⇒
256
y
1
2
=1−
400
81
=
400
319
⇒y
1
2
=
400
256×319
⇒y
1
=
20
16
319
=
5
4
319
=
5
4
319
ft
∴ the height of the arch 9 ft from the right of the centre is =
5
4
319
ft.
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