An arc of a bridge is in shape of a semi ellipse it is 8m wide and 2m height at the center (i) find the equation of arc , (ii)Distance between the two end points of the arc , (iii) center of the arc , (iv)height of the arc at the end points.
Answers
Step-by-step explanation:
ANSWER
Since the height and width of the arc from the centre is 2 m and 8 m respectively. It is clear that the length of the major axis is 8 m, while the length of the semi-minor axis is 2 m
The origin of the coordinate plane is taken as the centre of the ellipse while the major axis is taken along the x-axis. Hence the semi-ellipse can be diagrammatically represented as,
The equation of the semi-ellipse will be of the form
a
2
x
2
+
b
2
y
2
=1,y≥0 where a is the semi-major axis.
Accordingly, 2a=8
a=4,b=2
Therefore, the equation of the semi-ellipse is
16
x
2
+
4
y
2
=1,y≥0 ...(1)
Let B be a point on the major axis such that AB=1.5 m
Draw BC ⊥ OA
OB=(4−1.5) m =2.5 m
The x-coordinate of point C is −2.5 m.
On substituting the value of x with −2.5 in equation (1), we obtain
16
(−2.5)
2
+
4
y
2
=1
⇒
16
6.25
+
4
y
2
=1
⇒y
2
=4(1−
16
6.25
)
⇒y
2
=4(
16
9.75
)
⇒y
2
=2.4375
⇒y=1.56 (approx) (∵y≥0)
∴AC=1.56 m
Thus the height of the arch at a point 1.5 m from one end is approximately 1.56 m.