Driveway Problem

[mattholt12]

Hi, this is a real world problem. I'm 45 yrs old and trying to re-learn maths. I've encountered this real world problem and I just can't solve it. I'm getting lost in formulas and getting nowhere. This is a good little problem but has me perplexed.

The problem is: Given that a driveway crossing angle for a sagged crossing can have a minimum angle of 168 degrees (theta) between road and driveway (which is roughly a change in gradient of 9%). I should be able to work out a general formula that equates change in gradient with theta. ie if you change theta then you get a change in gradient without knowing anything else. I just can't seem to do it. It is simple trigonometry but I'm stumped. I need help.



I've tried to represent the problem as two gradients and tried to cancel everything out except delta g (change in gradient) and theta.

I know

x1/y1 = g1 (gradient 1)

x2/y2 = g2 (gradient 2)

Delta G = g2 - g1 (assume absolute values for calcs)

180deg = theta + alpha + Beta

Tan(alpha) = g1
Tan (Beta) = g2

Delta G = g2-g1 = Tan(Beta) - Tan(Alpha)

Delta G = 0.09

Tan(Beta) - Tan(Alpha) = 0.09

When Delta G = 0.09, what is Theta?

I want to have general Trig formula that works out Theta if you know Delta G or vice versa.