Solution:

Given data:

Rising grade = +0.80

Falling grade = -0.70

Rate of change of grade = 0.05 per chain

Calculation:

1) Determine the length of the vertical curve using the formula:

L = (Kv+Kp)S

Where,

L = Length of the vertical curve

Kv = Vertical curve gradient

Kp = Vertical curve length correction factor

S = Horizontal distance between the two points of the vertical curve

2) Calculate Kv:

Kv = (G2-G1)/2C

Where,

G1 = Initial grade

G2 = Final grade

C = Rate of change of grade

Kv = (0.80 - (-0.70)) / (2 x 0.05) = 15

3) Calculate Kp:

Kp = [((G1^2)+(G2^2))/(24C)] x (S^3)

Kp = [((0.80^2) + ((-0.70)^2)) / (24 x 0.05)] x (S^3) = 0.41S^3

4) Substitute the values of Kv and Kp in the formula to calculate the length of the vertical curve:

L = (Kv+Kp)S

L = (15 + 0.41S^3)S

The length of the vertical curve is (15 + 0.41S^3)S.

Explanation:

A parabolic vertical curve is a curve that is used to connect two grades in a road or railway. In the given problem, we are given the rising grade, falling grade, and the rate of change of grade. We need to calculate the length of the vertical curve.

To calculate the length of the vertical curve, we use the formula L = (Kv+Kp)S, where Kv is the vertical curve gradient, Kp is the vertical curve length correction factor, and S is the horizontal distance between the two points of the vertical curve.

We first calculate Kv using the formula Kv = (G2-G1)/2C, where G1 is the initial grade, G2 is the final grade, and C is the rate of change of grade. We then calculate Kp using the formula Kp = [((G1^2)+(G2^2))/(24C)] x (S^3). Finally, we substitute the values of Kv and Kp in the formula L = (Kv+Kp)S to get the length of the vertical curve.

In this problem, we get the length of the vertical curve as (15 + 0.41S^3)S.

0.8 - (-0.7) = 1.5%
1.5/ 0.05 = 30 chains