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CALCULUS

Suppose that waste from a factory began contaminaning a pond at the time of a spill in the factory. The pond has 50,000 gallons of water. Let Q(t) be the amount of contaminant in the pond t hours after the spill took place and is measured in pounds. The rate at which the contaminant is entering the lake is given by the formula 100e^(-0.05t) lb/hout, which decreases as time goes on. Clean water enters the lake at a rate of 1000 gallons/hour; the (contaminated) water leaves the lake at the same rate. Determine the differenitial equation for Q(t). What is IVP? Approximately what do you expect the contamination level to be after many days? (Note that you are not being asked to solve the equation or IVP).

Please show all steps and an explation. Thank you!

LET THE AMOUNT OF CONTAMINANT IN THE pond AT ANY TIME T BE Q
AND LET IT CHANGE BY AN AMOUNT DQ
IN SMALL PERIOD OF TIME DT.
ORIGINAL SOLUTION IN POND AT T=0 ......= 50000 G.
QTY OF CONTAMINANT ORIGINALLY AT T=0............=
RATE OF INPUT OF CONTAMINANT= 100[E^(-0.05T)] LB/HR
INFLOW RATE OF CLEAN WATER = 1000 GPH
OUT FLOW RATE = 1000 GPH
HENCE ACCUMULATION RATE = 0 GPH
HENCE SOLUTION AT ANY TIME T=50000 ...G.
CONTAMINANT BALANCE
INPUT
CONTAMINANT IN INFLOW=			100[E^(-0.05T)]* DT LB
OUT PUT
CONTAMINANT CONTENT IN OUT FLOW = Q/50000 LB/G
OUT PUT OF CONTAMINANT DURING DT = 1000*DT*Q/[50000] LB.
ACCUMULATION DURING DT = DQ = INPUT - OUT PUT
DQ =	100[E^(-0.05T)] DT -		0.02*QDT
[DQ/DT]+ 0.02 Q =		100[E^(-0.05T)]…………………………………1
EQN. 1 IS THE REQD. D.E. ………………..ANSWER		IS THE REQD. D.E. ………………..ANSWER
THE I.V.P. IS GIVEN BY EQN. 1 ABOVE & THE INITIAL CONDITION THAT
AT T=0 ……Q = 0
AS T TENDS TO INFINITY THE INPUT RATE OF CONTAMINANT
100[E^(-0.05T)] TENDS TO 100…..HENCE THE CONTAMINATION
LEVEL WILL BE 100 / 50000 = 1 / 500 LB / G