A correct solutions was also received from Ray Kremer.  Other correct solutions were received from Tim Kelley, Mariano A C de Andrade, Burkart Venzke, Monty Gray, Javier Echavarri, Robert McQuaid, Jack Wang, William Webb. A number of incorrect solutions were submitted.

Nathan Pauli's solution is a gem of simplicity.  Here it is in its entirety.

"I put the origin of my coordinate system at sea level, directly below the point where the 5% grade meets the parabola.

Because a parabola has a constant change in slope, the slope would be zero at the point that is 5/8'th of the way from the 5% grade to the 3% grade, which is 625 feet.

If we write the equation of the parabola in the form f (x) = a(x - b)² + c, then we've already found b = 625.

We can differentiate to find that f '(x) = 2a(x - 625).  We know that f '(0) = -.05, so we can solve for a and find that a = .00004.  Now we just have to plug in our one known point (0, 1250) to solve for c.

(a)  That gives the equation

(b)  f (1000) = the elevation where the parabola meets the 3% grade = 1240 ft.

(c) The drain should be at the lowest point on the parabola, which has location (625, 1234). "

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Page last updated 16 February 1999.