Question: The forward tangent (V-PT) and back tangent (V-PC) of a simple curve have azimuth $300^\circ$ and bearing $N\ 04^\circ\ E$, respectively. A third tangent line AB intersects the two tangent lines at bearing $S\ 34^\circ\ E$ – intersecting the back tangent at point A, and the forward tangent at point B. If the stationing of the vertex (V) of the simple curve is $16+464.35$ and the distance from point B to V of the curve is $277.6$ m, solve for the following: a. Radius of the simple curve that shall be tangent to these three lines. b. Stationing of the PC c. Stationing of the PT
1. **State the problem:**
We have a simple curve with vertex $V$ and two tangent lines: back tangent (V-PC) with bearing $N\ 04^\circ\ E$ and forward tangent (V-PT) with azimuth $300^\circ$. A third tangent line $AB$ intersects these tangents at points $A$ and $B$ with bearing $S\ 34^\circ\ E$. Given stationing of $V$ is $16+464.35$ and distance $BV=277.6$ m, find:
a. Radius $R$ of the simple curve tangent to these lines.
b. Stationing of PC.
c. Stationing of PT.
2. **Convert bearings and azimuths to angles from the north:**
- Back tangent bearing $N\ 04^\circ\ E$ means azimuth $4^\circ$.
- Forward tangent azimuth is $300^\circ$.
- Line $AB$ bearing $S\ 34^\circ\ E$ means azimuth $180^\circ + 34^\circ = 214^\circ$.
3. **Calculate angle between back and forward tangents:**
$$\theta = 300^\circ - 4^\circ = 296^\circ$$
Since angles in curves are usually less than $180^\circ$, take the smaller angle:
$$\theta = 360^\circ - 296^\circ = 64^\circ$$
4. **Calculate angle between back tangent and line AB:**
$$\alpha = 214^\circ - 4^\circ = 210^\circ$$
Smaller angle:
$$\alpha = 360^\circ - 210^\circ = 150^\circ$$
5. **Calculate angle between forward tangent and line AB:**
$$\beta = 300^\circ - 214^\circ = 86^\circ$$
6. **Use the Law of Sines in triangle VAB:**
Let $VA = x$, $VB = 277.6$ m (given), and angle at $V$ is $\theta = 64^\circ$.
The angles at $A$ and $B$ are $\alpha = 150^\circ$ and $\beta = 86^\circ$ respectively.
By Law of Sines:
$$\frac{x}{\sin(\beta)} = \frac{277.6}{\sin(\alpha)} = \frac{AB}{\sin(\theta)}$$
Calculate $x$:
$$x = \frac{\sin(86^\circ)}{\sin(150^\circ)} \times 277.6$$
Calculate values:
$$\sin(86^\circ) \approx 0.9976, \quad \sin(150^\circ) = 0.5$$
$$x = \frac{0.9976}{0.5} \times 277.6 = 1.9952 \times 277.6 = 554.1\ \text{m}$$
7. **Calculate length AB:**
$$AB = \frac{\sin(64^\circ)}{\sin(150^\circ)} \times 277.6$$
$$\sin(64^\circ) \approx 0.8988$$
$$AB = \frac{0.8988}{0.5} \times 277.6 = 1.7976 \times 277.6 = 498.8\ \text{m}$$
8. **Calculate radius $R$ of the simple curve:**
The radius formula for a simple curve is:
$$R = \frac{AB}{2 \sin(\theta/2)}$$
Calculate $\sin(\theta/2)$:
$$\theta/2 = 32^\circ, \quad \sin(32^\circ) \approx 0.5299$$
Calculate $R$:
$$R = \frac{498.8}{2 \times 0.5299} = \frac{498.8}{1.0598} = 470.4\ \text{m}$$
9. **Calculate stationing of PC:**
Stationing of $V$ is $16+464.35 = 16464.35$ m.
Distance $VA = 554.1$ m (from step 6).
Since PC is before $V$ along back tangent:
$$\text{Stationing of PC} = 16464.35 - 554.1 = 15910.25$$
10. **Calculate stationing of PT:**
Distance $VB = 277.6$ m (given).
Since PT is after $V$ along forward tangent:
$$\text{Stationing of PT} = 16464.35 + 277.6 = 16741.95$$
**Final answers:**
- a. Radius $R = 470.4$ m
- b. Stationing of PC = $15910.25$
- c. Stationing of PT = $16741.95$