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:
dC/dt + U dC/dx = K d^2C/dx^2
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 x = 0, the concentration can be shown—using some of the math needed for Cody Problem 51625—to be
C = (M/(A sqrt(4 pi K t))) exp(-(x-U t)^2/(4 K t))
Write a function to compute the length of stream affected by the spill. In other words, find the position x = L_a (say) beyond which the concentration never exceeds a threshold C = C_t.
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