The arch of a bridge is in the form of an arc of a circle. If the span of the bridge is 40 m and height in the middle is 8 m, then what is the radius of curvature of the bridge?
29 m
Understanding the Bridge Arch Problem
The question asks us to find the radius of curvature of a bridge arch. We are told the arch is shaped like an arc of a circle. We are given the span of the bridge, which is the total width across the bottom of the arch, and the height of the arch in the middle.
Defining the Geometry
Let's visualize the bridge arch as a segment of a circle. The span of the bridge is a chord of this circle. The height of the arch in the middle is the perpendicular distance from the midpoint of the chord to the highest point on the arc. This height is also known as the sagitta.
- Span of the bridge (\(S\)) = 40 m
- Height of the arch (\(h\)) = 8 m
- Radius of curvature (radius of the circle, \(r\)) = ?
The span (\(S\)) is the length of the chord. Half the span is the distance from the center of the chord to one end. Let's call half the span \(w\).
\[ w = \frac{S}{2} = \frac{40 \text{ m}}{2} = 20 \text{ m} \]
Imagine the center of the circle from which the arc is formed. The radius \(r\) is the distance from this center to any point on the arc. The highest point of the arch is on the arc, and its distance from the center is \(r\). The midpoint of the chord is located directly below the highest point. The distance from the center of the circle to the midpoint of the chord can be represented as \(r - h\).
Applying the Pythagorean Theorem
Consider a right-angled triangle formed by:
- The radius \(r\) as the hypotenuse (from the center of the circle to one end of the chord).
- Half the span \(w\) as one leg (from the midpoint of the chord to one end).
- The distance from the center of the circle to the midpoint of the chord (\(r - h\)) as the other leg.
According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
\[ (\text{Distance from center to chord midpoint})^2 + (\text{Half span})^2 = (\text{Radius})^2 \]
\[ (r - h)^2 + w^2 = r^2 \]
Calculating the Radius of Curvature
Now, we substitute the given values into the equation:
- \(w = 20\) m
- \(h = 8\) m
\[ (r - 8)^2 + (20)^2 = r^2 \]
Expand the term \((r - 8)^2\):
\[ r^2 - 2(r)(8) + 8^2 + 20^2 = r^2 \]
\[ r^2 - 16r + 64 + 400 = r^2 \]
\[ r^2 - 16r + 464 = r^2 \]
Subtract \(r^2\) from both sides of the equation:
\[ -16r + 464 = 0 \]
Add \(16r\) to both sides:
\[ 464 = 16r \]
Now, solve for \(r\) by dividing both sides by 16:
\[ r = \frac{464}{16} \]
Performing the division:
\[ r = 29 \]
So, the radius of curvature of the bridge arch is 29 meters.
Conclusion
Using the geometry of a circle and the Pythagorean theorem, we found that the radius of the circular arc that forms the bridge arch is 29 meters, given a span of 40 m and a height of 8 m.
| Parameter | Value | Source |
|---|---|---|
| Span (\(S\)) | 40 m | Given |
| Height (\(h\)) | 8 m | Given |
| Half Span (\(w = S/2\)) | 20 m | Calculated |
| Formula Used | \((r-h)^2 + w^2 = r^2\) | Pythagorean Theorem |
| Calculated Radius (\(r\)) | 29 m | Result |
Revision Table: Bridge Arch Radius Calculation
| Term | Definition | Relevance to Problem |
|---|---|---|
| Span | Horizontal distance covered by the arch; length of the chord. | \(S = 40\) m, used to find half-chord \(w\). |
| Height (Sagitta) | Vertical distance from the midpoint of the chord to the arc's peak. | \(h = 8\) m, used in the Pythagorean equation. |
| Radius of Curvature | Radius of the circle of which the arch is an arc. | The unknown \(r\) we need to find. |
| Chord | A line segment connecting two points on a circle (the span). | The span of the bridge is the chord. |
Additional Information: Circle Segments and Arches
The problem deals with a circular segment, which is a region of a circle cut off from the rest by a secant or chord. The arch itself is the circular arc bounding the segment.
There's a direct formula relating the radius (\(r\)), half-chord length (\(w\)), and sagitta (height, \(h\)) of a circular segment, derived from the same Pythagorean theorem used above:
\[ r = \frac{w^2 + h^2}{2h} \]
Let's check this formula with our values:
\[ r = \frac{(20 \text{ m})^2 + (8 \text{ m})^2}{2 \times 8 \text{ m}} \]
\[ r = \frac{400 \text{ m}^2 + 64 \text{ m}^2}{16 \text{ m}} \]
\[ r = \frac{464 \text{ m}^2}{16 \text{ m}} \]
\[ r = 29 \text{ m} \]
This confirms our result obtained using the step-by-step Pythagorean theorem approach. Understanding the geometry and how the radius, chord, and height relate in a circular segment is key to solving problems like this bridge arch calculation.
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