Question

A lake containing 100 million gallons of fresh water has a stream flowing through it. Water enters the lake at a constant rate of 5 million gal/day and leaves at the same rate. An upstream manufacturer begins to discharge pollutants into the feeder stream. Each day, during the from 6 a.m. to 6 p.m., the stream has a pollutant concentration of 3 mg/gal (3 x 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. hours a. 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 $q(t)$: where $ q'(t) = \begin{cases} \\ \\ \end{cases} $ and $c_q(t^+) - \\ $b. Take the Laplace transform of both sides of the differential equation formulated in part (a) to determine $Q(s) = \mathcal{L}\{q(t)\}$. $Q(s) = \mathcal{L}\{q(t)\} = $ $q(0) = $

          A lake containing 100 million gallons of fresh water has a stream flowing through it. Water enters the lake at a constant rate of 5 million gal/day and leaves at the same rate. An upstream manufacturer begins to discharge pollutants into the feeder stream. Each day, during the
from 6 a.m. to 6 p.m., the stream has a pollutant concentration of 3 mg/gal (3 x 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.
hours
a. 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 $q(t)$:
where
$
 q'(t) = \begin{cases}
 \\
 \\
 \end{cases}
$
and $c_q(t^+) - \\
$b. Take the Laplace transform of both sides of the differential equation formulated in part (a) to determine $Q(s) = \mathcal{L}\{q(t)\}$.
$Q(s) = \mathcal{L}\{q(t)\} = 
$
$q(0) = $

$A lake containing 100 million gallons of fresh water has a stream flowing through it. Water enters the lake at a constant rate of 5 million gal/day and leaves at the same rate. An upstream manufacturer begins to discharge pollutants into the feeder stream. Each day, during the from 6 a.m. to 6 p.m., the stream has a pollutant concentration of 3 mg/gal (3 x 10^-6 kg/gal); at other times, the stream feeds in fresh water. Assume that a well-stirred mixture leaves the lake and that the manufacturer operates seven days per week. hours a. Let t = 0 denote the instant that pollutants first enter the lake. Let q(t) denote the amount of pollutant (in kilograms) present in the lake at time t (in days). Use a c̈onservation of pollutantp̈rinciple (rate of change = rate in - rate out) to formulate the initial value problem satisfied by q(t): where q'(t) = and cq(t^+) -b. Take the Laplace transform of both sides of the differential equation formulated in part (a) to determine Q(s) = ℒ{q(t)}. Q(s) = ℒ{q(t)} = q(0) =$

Added by Gregorio L.

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