A vertical parabolic sag curve is to be designed to connect a down-gradient of 1 in 20 with an up-gradient of 1 in 15, the chainage and reduced level of the intersection point of the two gradients being 797.7 m and 83.544 m respectively. In order to allow for necessary headroom, the reduced level of the curve at chainage 788.7 m on the down-gradient side of the intersection point is to be 85.044 m. Calculate: (a) The reduced levels and chainages of the tangent points and the lowest point on the curve. (b) The reduced levels of the first two pegs on the curve, the pegs being set at the 30-m points of through chainage. (ICE) (Answer:(a)T1 = 745.24 m, 86.166 m, T2 = 850.16 m, 87.042 m, lowest pt = 790.21 m, 85.041 m, (b) 85.941 m, 85.104 m)

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To calculate the reduced levels and chainages of the tangent points and the lowest point on the curve, we can use the principles of vertical alignment design. Here's how you can calculate these values:

Step 1: Calculate the gradients between the intersection point and the tangent points.
- The down-gradient is 1 in 20, which means for every 20 units in the horizontal distance, there is a 1-unit rise in elevation.
- The up-gradient is 1 in 15, which means for every 15 units in the horizontal distance, there is a 1-unit rise in elevation.

Step 2: Determine the length of the curve.
- The chainage of the intersection point is 797.7 m.
- The chainage of the tangent point on the down-gradient side is 788.7 m.
- The chainage of the tangent point on the up-gradient side is unknown (T2).
- The length of the curve can be calculated as the difference between these chainages: Curve length = T2 - 788.7 m.

Step 3: Determine the elevation difference between the intersection point and the tangent points.
- The reduced level of the intersection point is 83.544 m.
- The reduced level of the tangent point on the down-gradient side is 85.044 m.
- The reduced level of the tangent point on the up-gradient side is unknown (T2).

Step 4: Determine the equations for the curve.
- The equation for a vertical parabolic sag curve can be written as: E = Ax^2 + Bx + C, where E represents the reduced level and x represents the chainage.

Step 5: Use the given values to solve for the coefficients A, B, and C.
- Using the coordinates of the intersection point (797.7 m, 83.544 m) and the tangent point on the down-gradient side (788.7 m, 85.044 m), we can set up two equations:
- 83.544 = A(797.7)^2 + B(797.7) + C
- 85.044 = A(788.7)^2 + B(788.7) + C

- By solving this system of equations, you can find the values of A, B, and C.

Step 6: Calculate the reduced level and chainage of the tangent points.
- Substitute the unknown chainage values (788.7 m and T2) into the parabolic curve equation to find the corresponding reduced levels.

Step 7: Calculate the reduced level and chainage of the lowest point on the curve.
- The lowest point on the curve occurs when the curve changes from sag to crest.
- Calculate the chainage and reduced level of this point using the equations: B^2 - 4AC = 0 and x = -B/(2A).

To calculate the reduced levels of the first two pegs on the curve at 30-meter intervals, you can use the same equation (E = Ax^2 + Bx + C) to calculate the reduced level for each specific chainage value. Substitute the chainage value into the equation to find the corresponding reduced level.

By following these steps, you should be able to obtain the answers provided in the question.