A pollutant enters a lake at a rate modeled by the differentiable function r, where r(t)=t2−6t+5, measured in milligrams per second, for 0≤t≤6 seconds.
Let M(t) denote the total mass (in milligrams) of the pollutant that has entered the lake from time t=0 up to time t, so that M(0)=0 and M′(t)=r(t).
(a) Find an explicit expression for M(t) for 0≤t≤6. Show all work and include units for each term in your antiderivative.
(b) Determine the time t in the interval 0≤t≤6 at which M(t) attains its maximum value. Justify your answer by analyzing the relationship between M and r.
(c) Compute the exact total mass of pollutant that enters the lake during the time interval 2≤t≤5 seconds. Show your work and include units.
(d) On which, if any, subintervals of [0,6] is r(t) negative? Explain how negative values of r(t) affect the behavior of M(t) and interpret this result in the context of pollutant flow into the lake.