1. Applications of Parabolas: Highway Overpasses using Type 1 Vertical Curves John Catlett Mathematics Teacher North Star High School

2. What is a parabola? A parabola is the graph that results from an equation of the form: Parabolas are symmetric about a vertical line known as the Axis of Symmetry. The Axis of Symmetry runs through the maximum or minimum point of the parabola which is called the Vertex.

3. Examples of parabolas:

4. Exploration of Parabolas How do the coefficients A, B and C affect the graph of the parabola? Let’s use Geogebra to find out: http://isite.lps.org/dtravis2/jcat/geogebra/par abolaexploration.html http://isite.lps.org/dtravis2/jcat/geogebra/par abolaexploration.html

5. Parabolas and Vertical Curves In road design, vertical curves are designed using parabolas. There are four types:

6. Focus: Type 1 Curves and Overpasses

7. Design Factors Parabolas used in the overpass design are based on a number of factors: – The entrance and exit grades, g 1 and g 2 – The length of the vertical curve, L (NDOR uses 600 ft as the minimum L) – The design speed (speed the road is designed for) – The sight stopping distance, S (based on design speed and line of sight) – The elevation at the start of the curve, Elev BVC

8. Equation of the parabola:

9. Finding the Parabola for Overpass Design

10. Lets find equations to fit some existing overpasses. http://isite.lps.org/dtravis2/jcat/geogebra/ove rpassexample1.html http://isite.lps.org/dtravis2/jcat/geogebra/ove rpassexample1.html For our purposes today, we will use: – The NDOR minimum ( 600 ft ) for L – The distance above the roadway that runs below the overpass at the start of the curve as the Elev BVC (these calculations are more complex in real life)

11. Now lets find the equation of the parabola using our formula,

12. Example: The entrance grade, g 1, is 3%, the exit grade, g 2, Is 2%, the Elev BVC is 8 feet and the horizontal length of the curve, L, is 900 feet. Solution: Equation: Check Solution: http://isite.lps.org/dtravis2/jcat/geogebra/overpassequationc hecker.html http://isite.lps.org/dtravis2/jcat/geogebra/overpassequationc hecker.html

13. Final Thoughts Video: http://isite.lps.org/dtravis2/jcat/http://isite.lps.org/dtravis2/jcat/ Other things to consider?