The offset distance of a simple curve from the PT to the tangent line passing through the PC is equal to 120 m. The simple curve has an angle of intersection of 50°. Find the radius of the simple curve.

To solve for the radius \( R \) of the simple curve, we can use the following relationship between the offset distance \( E \), the radius \( R \), and the angle of intersection \( I \):

\[
E = R \cdot (1 - \cos(I/2))
\]

Where:
- \( E \) is the offset distance from the PT to the tangent line passing through the PC (given as 120 m).
- \( I \) is the angle of intersection (given as 50°).

We need to find the radius \( R \).

First, let's express \( I/2 \) in radians:

\[
I/2 = 50^\circ / 2 = 25^\circ
\]

Convert \( 25^\circ \) to radians:

\[
25^\circ = \frac{25 \times \pi}{180} \approx 0.4363 \text{ radians}
\]

Now, we can substitute into the equation:

\[
E = R \cdot (1 - \cos(0.4363))
\]

Given that \( E = 120 \, \text{m} \), we solve for \( R \):

\[
120 = R \cdot (1 - \cos(0.4363))
\]

Calculate \( \cos(0.4363) \):

\[
\cos(0.4363) \approx 0.905
\]

Substitute this value:

\[
120 = R \cdot (1 - 0.905)
\]

\[
120 = R \cdot 0.095
\]

Finally, solve for \( R \):

\[
R = \frac{120}{0.095} \approx 1263.16 \, \text{m}
\]

So, the radius \( R \) of the simple curve is approximately **1263.16 meters**.

Would you like any further details or have any questions? Here are some related questions you might consider:

1. How is the offset distance related to the radius in the case of a circular curve?
2. What are the other ways to calculate the radius of a curve if the length of the curve is known?
3. How does the angle of intersection affect the radius of a simple curve?
4. Can the same formula be used for non-circular curves?
5. How do different units (degrees vs. radians) impact the accuracy of curve-related calculations?

**Tip:** Always convert angles to radians when using trigonometric functions in formulas, as many calculators and formulas assume angles are in radians.

Discover how to find the radius of a simple curve with an angle of intersection of 50° and an offset distance of 120 m. This mathematical problem involves trigonometry and geometric constructions.

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