Concept explainers
To find: the height of the arch above the water
Answer to Problem 27WE
The height of the arch above the water is 3 m.
Explanation of Solution
Given:
The span of the bridge is 12m
Height is 4m
Calculation:
Let us consider a semi-ellipse with the major axis horizontal, focal radii
The span of the bridge is the length of the major axis,
The height of the ellipse, b , is the height of the arch above the water at the center, which is 4 m.
Since the major axis is horizontal, the foci are on the x -axis.
The ellipse having center
Substitute 6 for a , and 4 for b in the equation to get the equation of the ellipse.
Now, consider a point on the major axis, 2m from one of the ends of the ellipse.
The distance of this point from the center of the ellipse is
Find the height of the arch from the point
For this, consider a point vertically above
Let h be the height of the arch above this point.
The coordinates of the point on the arch is
Substitute 4 for x , and h for y in the equation of the ellipse and simplify.
Take the square root on both the sides.
Conclusion:
Therefore, the height of the arch above the water is 3 m.
Chapter 9 Solutions
Algebra and Trigonometry: Structure and Method, Book 2
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