Question

(Modeling) Length of a Sag Curve When a highway goes downhill and then uphill, it has a sag curve. Sag curves are designed so that at night, headlights shine sufficiently far down the road to allow a safe stopping distance. See the figure. S and L are in feet. The minimum length L of a sag curve is determined by the height h of the car's headlights above the pavement, the downhill grade θ₁  0°, and the safe stopping distance S for a given speed limit. In addition, L is dependent on the vertical alignment of the headlights. Headlights are usually pointed upward at a slight angle α above the horizontal of the car. Using these quantities, for a 55 mph speed limit, L can be modeled by the formula (θ₂ - θ₁)S² L = ————————— , 200(h + S tan α) where S \u003c L. (Data from Mannering, F., and W. Kilareski, Principles of Highway Engineering and Traffic Analysis, Second Edition, John Wiley and Sons.) Compute length L, to the nearest foot, if h = 1.9 ft, α = 0.9°, θ₁ = -3°, θ₂ = 4°, and S = 336 ft.

Accepted Answer

Identify the given values from the problem: height of headlights $$h = 1.9 ft$$, angle of headlights above horizontal $$\alpha = 0.9^\circ$$, downhill grade $$\$$theta_1$$ = -3^\circ$$, uphill grade $$\$$theta_2$$ = 4^\circ$$, and safe stopping distance $$S = 336 ft. $$Recall the formula for the length $$L of $$the sag curve:

$$L = \frac{(\$$theta_2$$ - \$$theta_1$$) S^2$}{200 (h + S 	an \alpha)}$$

where all angles are in degrees and $$S \u003c L. $$Calculate the difference in grades:

$$\$$theta_2$$ - \$$theta_1$$ = 4^\circ - (-3^\circ) = 7^\circ. $$Compute the tangent of the angle $$\alpha$$:

$$	an \alpha = 	an 0.9^\circ ($$make sure your calculator is in degree mode). Substitute all known values into the formula and simplify step-by-step:

$$L = \frac{7 	imes (336)^2$}{200 	imes (1.9 + 336 	imes 	an 0.9^\circ)}$$

Evaluate the numerator and denominator separately before dividing to find $$L.$$

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Key Concepts

Trigonometric Functions and Angle Measures

Sag Curve Geometry and Vertical Alignment

Application of the Given Formula for Length L