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.