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# s53 - If we write the equation of the parabola in the form...

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Solution to Problem 53 Congratulations to this week’s winner Nathan Pauli 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.

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Unformatted text preview: 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) = 2 a ( 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 f ( x ) = .00004( x- 625) 2 + 1234.375 (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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s53 - If we write the equation of the parabola in the form...

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