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Calculus Highway Construction Problem

  1. Jan 17, 2008 #1
    1. The problem statement, all variables and given/known data

    A segment of interstate highwya under consideration lies in a vertical plane. A segment with a grade of -5% is to be joined smoothly to a segment with a grade of 3%. The vetical parabolic curve which will accomplish this transition is to begin at a point with elevation 1250 ft. The length of the vertical curve is to be 1000 ft. See figure. I have attached the figure as an attachment.

    Using a suitable coordinate system in which the horizontal and vertical units are feet, determine the equation of the vertical parabolic curve which makes the smooth transition between the two grades.

    2. Relevant equations

    y = Ax^2+Bx+C

    Grade of 3% means for every 100 ft. in highway length, there is a 3 ft. increase in elevation of the highway.
    Same for -5%.

    3. The attempt at a solution

    What I think is necessary to solve this problem is the coordinate point at B. Also, where the parabolic curve has a tangent of 0. With that I think I would be able to solve the problem.

    However, I can't figure out how to find where the parabolic curve has a tangent of 0.

    I may be on the wrong path completely so please let me know if you think there's a better way to solve the problem.

    Another thing I tried was saying that well, if it begins as a -5% grade, and ends as a 3% grade, the overall grade is -2%, thus from that I thought I could find the y coordinate point of B, but that got me nowhere I think.

    I would really appreciate it if anyone could help.
    Thanks a lot.

    Attached Files:

  2. jcsd
  3. Feb 1, 2008 #2
    You've recognised the general equation of a parabola to be y = Ax^2 + Bx + C. Because we're dealing with gradients, I suggest you differentiate this equation with respect to x. In this situation, y is the elevation, and x is the horizontal distance.

    A sensible co-ordinate system would be to have x = 0 when the parabola starts, i.e. y = 1250 when x = 0. Find C from this. Use the gradient when x = 0 to find B, and then also you known when x = 1000, the gradient is 3%, and so find A. Remember to turn the percentage gradients into fractions.

    Hope this helps.
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