Question

Integration by separation of variables: algebraic rearrangement?

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))`

Answer 1

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

Explanation:

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))`