When a contaminant is spilled into a stream, one might want to know how much of the stream is affected—e.g., the length over which the concentration exceeds a specified threshold. The concentration C is often computed as a function of time t and distance x from the spill using the advection-dispersion equation:
where U is the mean velocity of the river and K is a dispersion coefficient, which describes spreading by several mechanisms. For an instantaneous spill of mass M mixed over the cross section (with area A) at 
, the concentration can be shown—using some of the math needed for Cody Problem 51625—to be
, the concentration can be shown—using some of the math needed for Cody Problem 51625—to be
Write a function to compute the length of stream affected by the spill. In other words, find the position 
(say) beyond which the concentration never exceeds a threshold 
.
(say) beyond which the concentration never exceeds a threshold . Solution Stats
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I feel compelled to point out that the precision I requested in the tests is unrealistic in practice. These lengths do not need to be known to the centimeter, and dispersion coefficients are usually good to a factor of 2.