On page 305 we briefly considered a solution to the diffusion equation, which models how a pollutant, for example, diffuses from a single concentrated release. If you measure the concenIn other words, the pollutant is dumped into the environment only once, and then allowed to diffuse from the dumping point, without any more being added. This is only a good model for pollution dispersal over very short distances. Over longer distances you need to take things like air and water currents into account.

tration,

c, of the pollutant at point x =

a, then a good model is c(t) =

e

−

a 2

4t

√

4πt

.

Calculate c 0

(t) and c 00(t) when a = 5. Plot c(t) and its derivatives, and show that c initially increases, then decreases. What is limt→0 c(t)? Does this limit depend on a? Explain why this makes scientific sense. Find the time at which c attains its maximum value.