a simple curve is to be run joining two tangents of bearings s76°50′w and n77°00′w and is to pass a point x of coordinates 43.88n and 11.26e. if the coordinates of pc are 50.00n and 50.00e and the stationing of vertex is 2+972.45, find the radius of the curve, the stationing of pt and the stationing of point x.

To solve this problem, we need to determine the radius of the curve, the stationing of the point of tangency (PT), and the stationing of point X based on the given coordinates and bearings.

### Step 1: Identify the Elements

1. **Bearings**:
   - Bearing 1 (Tangent 1): S76°50′W
   - Bearing 2 (Tangent 2): N77°00′W

2. **Coordinates**:
   - Point of curvature (PC): (50.00 N, 50.00 E) = (50, 50)
   - Point X: (43.88 N, 11.26 E) = (43.88, 11.26)

3. **Stationing**:
   - Vertex stationing: 2+972.45 m (or 2972.45 m)

### Step 2: Calculate the Intersection Angle

The angle between the two tangents can be calculated by converting the bearings into angles:

- **Bearing 1**: S76°50'W → 180° + 76.83° = 256.83°
- **Bearing 2**: N77°00'W → 360° - 77° = 283°

#### Angle of Intersection
\[
\theta = |283° - 256.83°| = 26.17°
\]

### Step 3: Determine the Radius of the Curve

To find the radius (R) of the curve, we can use the formula for the radius of a circular curve that is defined by the angle of intersection and the stationing at the vertex:

\[
R = \frac{L^2}{2 \cdot D}
\]

Where:
- L is the length of the curve
- D is the angle of intersection in radians: \(D = \theta \times \frac{\pi}{180}\)

To proceed, we need to find the length of the curve (L). This requires additional geometric information or design parameters which are typically provided.

### Step 4: Calculate Stationing of PT

The stationing of the PT can be found by considering the tangent length (T) from the PC to the PT. This length can be computed from the radius and the angle of intersection:

\[
T = R \cdot \tan\left(\frac{\theta}{2}\right)
\]

Then, the stationing of PT will be:
\[
\text{Stationing of PT} = \text{Stationing of PC} + R \cdot (1 - \cos\left(\frac{\theta}{2}\right))
\]

### Step 5: Calculate Stationing of Point X

To find the stationing of point X, we need to determine the distance from PC to X. This can be calculated using the distance formula:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Where:
- \( (x_1, y_1) \) = coordinates of PC
- \( (x_2, y_2) \) = coordinates of X

### Conclusion

For further calculations, we need either the curve length or a specific radius, or any additional data that relates the coordinates or curve properties. Please provide any additional information if available.

If you have any questions or need further clarification, feel free to ask!

### Related Questions
1. What is the length of the curve or radius you want to assume?
2. Do you need help converting bearings to angles?
3. Would you like to understand how to calculate the angle of intersection more thoroughly?
4. Is there any specific method you prefer for calculating the distances?
5. Do you want to know how to visualize this scenario graphically?

### Tip
When working with bearings and curves, always keep track of your angles in both degrees and radians to simplify calculations.

A simple curve is to be run joining two tangents of bearings s76°50′w and n77°00′w and is to pass a point x of coordinates 43.88n and 11.26e. If the coordinates of pc are 50.00n and 50.00e and the stationing of vertex is 2+972.45, find the radius of the curve, the stationing of pt and the stationing of point x.

Explore the process of calculating the radius of a curve and stationing points in road design. This problem involves bearings, coordinates, and trigonometric calculations, suitable for advanced geometry and trigonometry students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation