Question 2: Write the differential equation for concentration c(t) of pollution in lake at any

time t assuming that the lake has constant volume V with f as the rate at which water flows

in/out of the lake and cin is the concentration of the pollutant in the flow entering the lake.

Solve the equation with initial condition c(0)= a, assuming that f is constant.

Determine the concentration at time t when only fresh water flows into the lake.

Find the time taken for lake’s pollution level to reach 5% of its initial value if only fresh

waters flows into the lake.

Apply to two American Lakes :

Lake Erie for which V = 458*109

m3, f = 1.75*10

11 m3/year

Lake Ontario for which V = 1636*109

m3, f = 2.089*10

11m

3 year and

Lake Burley Griffin for which constant volume V = 28*10^6 m^3, f = 4*106

m3 �month

In[7]:= Clear@c, tDdiffeqtn = c'@tD � HF in - F c@tDL 1 � V

diffsol = DSolve@8diffeqtn, c@0D � a<, c@tD, tDfrshflweqn = Hc@tD �. diffsolL �. in ® 0

lakeerie = frshflweqn �. 8F ® 1.75 * 10^11, V ® 458.0 * 10^9<lakeontario = frshflweqn �. 8F ® 2.089 * 10^11, V ® 1636.0 * 10^9<lakeburleygriffin = frshflweqn �. 8F ® 4 * 10^6, V ® 28.0 * 10^6<sol = Solve@Part@frshflweqn, 1D � 0.05 * a, tD

Out[8]= c¢@tD �

F in - F c@tDV

Out[9]= ::c@tD ® ã-

F t

V Ka - in + ã

F t

V inO>>

Out[10]= :a ã-

F t

V >

Out[11]= 9a ã-0.382096 t=

Out[12]= 9a ã-0.127689 t=

Out[13]= 9a ã-0.142857 t=

Solve::ifun : Inverse functions are being used by Solve, so

some solutions may not be found; use Reduce for complete solution information. �

Out[14]= ::t ®2.99573 V

F

>>

In[15]:= Plot@8lakeerie �. a ® 10, lakeburleygriffin �. a ® 10, lakeontario �. a ® 10<,

8t, 0, 10<, PlotLegends ® 8"Erie", "Burley Griffin", "Ontario"<,

PlotLabel ® "Lakes' Pollution Models",

AxesLabel ® 8"Time in year", "Pollutant in Lake"<D

Out[15]=

2 4 6 8 10

Time in year

2

4

6

8

10

Pollutant in Lake

Lakes' PollutionModels

Erie

Burley Griffin

Ontario

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