The tangents of a simple curve have bearings of N 20° E and N 80° E, respectively. The radius of the curve is 200 m. Compute the external distance of the curve


To solve this problem, we need to compute the **external distance** (also known as the **external radius**) of a simple circular curve. The external distance is the distance from the **center of the curve** to the **tangent** at the point of intersection between the tangents.

### Step-by-step process:

1. **Understanding the Bearings:**
   The bearings of the tangents are given as:
   - First tangent: **N 20° E**
   - Second tangent: **N 80° E**

   The angle between the tangents is the difference in the bearings. Since both bearings are measured from the North, we can compute the angle between them by subtracting the smaller bearing from the larger one:

   \[
   \theta = 80^\circ - 20^\circ = 60^\circ
   \]
   
   So, the angle between the tangents is **60°**.

2. **Given Information:**
   - The **radius (R)** of the curve is 200 m.
   - The **angle between the tangents (θ)** is 60°.

3. **Formula for External Distance (E):**
   The external distance \( E \) can be computed using the following formula:

   \[
   E = R \cdot \cot\left(\frac{\theta}{2}\right)
   \]

   Where:
   - \( R \) is the radius of the curve (200 m).
   - \( \theta \) is the angle between the tangents (60°).

4. **Calculation:**

   \[
   E = 200 \cdot \cot\left(\frac{60^\circ}{2}\right) = 200 \cdot \cot(30^\circ)
   \]
   
   We know that \( \cot(30^\circ) = \frac{1}{\tan(30^\circ)} = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \).

   So,

   \[
   E = 200 \cdot \sqrt{3} \approx 200 \cdot 1.732 = 346.4 \, \text{m}
   \]

### Final Answer:
The external distance of the curve is approximately **346.4 meters**.

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Would you like a more detailed explanation of the formula or further clarifications?

Here are some related questions to expand your understanding:
1. How do you calculate the internal distance of a circular curve?
2. What is the relationship between the external distance and the length of the curve?
3. How does the angle between tangents affect the external distance?
4. What is the formula for computing the radius of a curve based on the external distance?
5. How can the tangent bearings help in identifying the curve’s center point?

**Tip:** To always check if you've used the correct angle between tangents, verify the bearings are referenced from the same point, typically North or South, to avoid confusion when calculating the angle.

The tangents of a simple curve have bearings of N 20° E and N 80° E, respectively. The radius of the curve is 200 m. Compute the external distance of the curve

Learn how to compute the external distance of a simple curve with tangents at bearings N 20° E and N 80° E. This problem involves using trigonometric formulas to determine the external radius of a circular curve. Suitable for high school students in Grades 11-12.

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