Integration by separation of variables: algebraic rearrangement?

Question

A lake contains 5,000,000 million litres of unpolluted water. A river flows into the lake at 100,000 litres per day. Due to polluters, the river flowing in contains 5 grams per litre of pollutant. A river flows out of the lake at 100,000 litres per day. Find an expression for the amount of pollutant in the lake.

I have:
#(dp)/dt = 500,000-p/50#
#p=25,000,000+e^(-t/50+c)#

Answer says #p=25,000,000(1-e^(-t/50))#

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Answer 1 · 2016-05-13T16:50:49

your answer is almost correct. Needs to get rid of c only, as explained below.

Your derivation is ok. Only thing left is to determine the constant of integration c.

For this apply the initial condition that at t=0, p=0 (there was no pollution initially)

Thus # 0= 25,000,000+e^c#

Thus #e^c= -25,000,000#. Your answer would then become

#p= 25,000,000+e^(-t/50) . e^c#

#p= 25,000,000-25,000,000 e^(-t/50)#

#=25,000,000(1-e^(-t/50))#