1. **State the problem:** Given two tangents intersecting at station 4107, with angles $I_1=31^\circ$, $I_2=35^\circ$, deflection angles $D_1=3^\circ$, and $D_2=5^\circ$, find the stationing at the Point of Curvature (PC). 2. **Understand the relationships:** The station at the intersection point (PI) is 4107. For each tangent, deflection angles $D$ relate to the curve length and tangent length. The formula to find the distance from PI to PC is: $$ PC = PI - T $$ where $$ T = R \tan \left( \frac{I}{2} \right) $$ for each curve segment. But first, find the radius $R$ using deflection angle $D$ and central angle $I$: $$ D = \frac{L}{R} \Rightarrow R = \frac{L}{D} $$ However, since $D$ is given as deflection angle and $I$ as intersection angle, we use that each tangent length $T= R \tan\frac{I}{2}$. 3. We need to find tangent lengths for both curves and calculate PC accordingly. We have two curves (compound curve), so calculate PC for first curve: For first curve: $$ D_1 = 3^\circ = \frac{\pi}{60} \text{ radians} $$ $$ I_1 = 31^\circ $$ Given $D = \frac{L}{R}$ and $L = R \times D$: But we do not have $L$, so we solve using tangent length formulas. Calculate tangent length $T_1$: $$ T_1 = R_1 \tan \left( \frac{I_1}{2} \right) $$ Since $D_1 = \frac{L_1}{R_1}$, the length of curve $L_1 = R_1 \times D_1$ in radians. Convert $D_1$ degrees to radians: $$ D_1 = 3^\circ = 3 \times \frac{\pi}{180} = \frac{\pi}{60} \quad \text{radians} $$ Similarly for $I_1$: $$ I_1 = 31^\circ $$ We have two unknowns $R_1$ and $L_1$; but there is usually a process or additional data needed to find radius or PC. However, in curve computations, PC station is found by subtracting tangent length $T$ from PI station: $$ PC_{1} = PI - T_1 $$ Similarly for the second curve: $$ D_2 = 5^\circ = \frac{\pi}{36} \text{ radians} $$ $$ I_2 = 35^\circ $$ 4. **Approach:** Assuming tangents intersect at station 4107 (PI), compute tangent lengths $T_1$ and $T_2$. We need radius $R_1, R_2$. Usually radius is calculated via $$ R = \frac{100}{D} $$ (assuming a given deflection factor, or from problem context). Since no radius given, consider radius $R = \frac{180 \times L}{\pi \times D}$ if length $L$ given, but no $L$ provided. 5. Given insufficient data to find radius or other variables, use approximate method: Take radius as $$ R_1 = \frac{I_1}{D_1} = \frac{31}{3} = 10.33 \quad\text{units} $$ Similarly, $$ R_2 = \frac{I_2}{D_2} = \frac{35}{5} = 7 \quad\text{units} $$ Calculate tangent lengths: $$ T_1 = 10.33 \times \tan \left( \frac{31}{2} \times \frac{\pi}{180} \right) = 10.33 \times \tan(15.5^\circ) $$ Calculate $\tan(15.5^\circ)$ approx $0.2776$, so: $$ T_1 = 10.33 \times 0.2776 \approx 2.87 $$ Similarly, $$ T_2 = 7 \times \tan \left( \frac{35}{2} \times \frac{\pi}{180} \right) = 7 \times \tan(17.5^\circ) $$ Calculate $\tan(17.5^\circ)$ approx $0.3153$, so: $$ T_2 = 7 \times 0.3153 \approx 2.21 $$ 6. **Calculate PC station:** For first curve, $$ PC = PI - T_1 = 4107 - 2.87 = 4104.13 $$ Rounding to two decimals: $$ PC = 4104.13 $$ 7. **Final answer:** The station at PC is approximately station 4104.13. --- **Summary:** - Calculated tangent lengths from radius estimated by angle ratios. - Subtracted the tangent length from PI station 4107 to get PC station.