I don't care for the question too much. To me, the term "offset" in the problem statement should be "Tangent Offset", or "Vertical Curve Tangent Offset". y' is typically used to represent the VC Tangent Offset. I also think when they give the benchmark station and elevation, it should be clear that the elevation is on the curve. What's with mixing "existing" curve (meaning actual/as-built) with "designed" entering grade (not necessarily actual/as-built)? The solution does not include units. And everyone knows x should be a little x, not a big X! Moving on....
There are two formulas to calc the VC Offset that I know of:
y' = (r/2)(x^2); r being (well, you know what r is) and x being the distance to the point on the curve measured from the BVC, in stations. y' is in feet and I believe always from the incoming tangent (g1) to the curve.
- or -
y' = (4 M x^2) / L^2; M being the Middle Ordinate Distance. M = |g1 - g2| L / r - or - M = (1/2)[(ELbvc + ELevc)/2 - ELpvi]
r = [(-4.7 - 3.6) / 7] x^2 = -1.1857 %/sta
M = |3.6 - (-4.7)| 7 / -1.1857 = 7.26 ft
x = 18+50 - 23+00 = 550 ft = 5.5 sta
y' = (r/2)x^2 = (-1.1857/2)(5.5)^2 = -17.93 ft
y' = 4 M x^2 / L^2 = (4)(7.26)(7^2) / 5.5^2 = 17.93 ft (or, -17.93 ft since g1 is above the curve)
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