A lake containing 80 million gallons of fresh water has a stream flowing through it. Water enters the lake at a constant rate of 4 million gal/day and leaves at the same rate. An upstream manufacturer begins to discharge pollutants into the feeder stream. Each day, during the hours from 6 a.m. to 6 p.m., the stream has a pollutant concentration of 3 mg/gal $(3 \times 10^{-6}$ kg/gal$)$; at other times, the stream feeds in fresh water. Assume that a well-stirred mixture leaves the lake and that the manufacturer operates seven days per week.

1. Let $t=0$ denote the instant that pollutants first enter the lake. Let $q(t)$ denote the amount of pollutant (in kilograms) present in the lake at time $t$ (in days). Use a "conservation of pollutant" principle (rate of change = rate in - rate out) to formulate the initial value problem satisfied by $q(t)$:
   
   $q' =$ $\cdot c_i \ - \ $ $\cdot q, \quad \quad q(0) =$
   where

   $c_i = \displaystyle \Bigg\lbrace$

   if $0 \leq t < \frac{1}{2},$ if $\frac{1}{2} \leq t < 1.$

   
   and $c_i(t +$ $) = c_i(t)$.
   
   
2. Take the Laplace transform of both sides of the differential equation formulated in part (a) to determine $Q(s) = {\mathcal L}\left\lbrace q(t) \right\rbrace$.
   
   $\displaystyle Q(s) = {\mathcal L}\left\lbrace q(t) \right\rbrace =$ help (formulas)