Problem: A symmetrical parabolic vertical curve has a length $L = 200\text{ m}$. The initial grade is $g_1 = +2\%$ and the final grade is $g_2 = -3\%$. The Point of Vertical Intersection (PVI) is located at station $20+00$ with an elevation of $100.00\text{ m}$. Calculate the station and elevation of the highest point on the curve.

Problem: A symmetrical parabolic vertical curve has a length $L = 150\text{ m}$. The initial grade is $g_1 = -4\%$ and the final grade is $g_2 = +2\%$. The Point of Vertical Curvature (PVC) is located at station $10+00$ with an elevation of $120.00\text{ m}$. What is the elevation of the Point of Vertical Tangency (PVT)?

Problem: For the same symmetrical vertical curve (Example 2), calculate the vertical offset $y$ at a horizontal distance $x = 50\text{ m}$ from the PVC.

Problem: An unsymmetrical parabolic vertical curve has a total length $L = 300\text{ m}$. The horizontal distance from the PVC to the PVI is $L_1 = 120\text{ m}$ and the distance from the PVI to the PVT is $L_2 = 180\text{ m}$. The initial grade is $g_1 = -2\%$ and the final grade is $g_2 = +3\%$. What is the external distance $E$ at the PVI?

Problem: A highway passing under a bridge requires an elevation of at least $105.00\text{ m}$ at the PVI station to maintain clearance. The PVC is at elevation $108.00\text{ m}$ and $L_1 = 80\text{ m}$. The initial grade is $-4\%$. What is the maximum length of $L_2$ if the final grade is $+2\%$ to ensure the sag curve clears the bridge constraints? The external distance must not raise the curve above $105.00\text{ m}$ at the PVI.

Problem: For an unsymmetrical curve with $L_1 = 150\text{ m}$, $L_2 = 100\text{ m}$, $g_1 = 3\%$, and $g_2 = -1\%$, calculate the difference in elevation between the initial tangent at the PVI and the curve itself at that same station.

Problem: A crest vertical curve must be designed for a stopping sight distance of $S = 140\text{ m}$. The initial grade is $g_1 = +4\%$ and the final grade is $g_2 = -2\%$. Assuming driver eye height $h_1 = 1.08\text{ m}$ and object height $h_2 = 0.60\text{ m}$, determine the required length of the curve. Assume $S < L$.

Problem: A sag vertical curve has a length $L = 250\text{ m}$. The grades are $g_1 = -3\%$ and $g_2 = +3\%$. The stopping sight distance required is $S = 180\text{ m}$. Determine if the curve provides adequate headlight sight distance. Use the standard formula for sag curves $L = \frac{A S^2}{120 + 3.5S}$ assuming $S < L$.

Problem: A minor crest vertical curve connects an initial grade of $g_1 = +2\%$ and a final grade of $g_2 = -1\%$. The required stopping sight distance is $S = 200\text{ m}$. Assuming driver eye height $h_1 = 1.08\text{ m}$ and object height $h_2 = 0.60\text{ m}$, determine the required length of the curve. Assume $S > L$.