Forming a Differential equation for a lake pollution model

[Nutty_Noodle]

The volume of lake = 150*10^9 Litre
The flow rate in and out of the lake = 20*10^6 Liter/year
Concentration, C(t)=4+sin(2t) grams/Litre
Assumptions:
Pollution not created or destroyed in the lake
Lake is well mixed

Form a differential equation describing this contamination flow and determine the amount Q(t) of toxic chemicals in the lake at time t

so yeah I am lost as to where to even start, I have researched other models but they all solve for C(t) so any and all help is welcomed, links or videos to read/watch as well.

thank you for your time

Noodle

 

[Cubist]

so yeah I am lost as to where to even start

Think about Q(t), the total grams of pollutant in the lake. What makes this amount change over time? It's worth spending a couple of minutes thinking about this for yourself before you read on...

How many grams of contaminant flow IN to the lake per unit time, via the incoming flow (which is polluted with concentration C(t) )?
How many grams of contaminant flow OUT of the lake per unit time? (this will be related to the current concentration in the lake, and the rate of water outflow)

Combine the two answers above to write dQ(t)/dt

 

[Nutty_Noodle]

How many grams of contaminant flow IN to the lake per unit time, via the incoming flow (which is polluted with concentration C(t) )?
How many grams of contaminant flow OUT of the lake per unit time? (this will be related to the current concentration in the lake, and the rate of water outflow)

so would C(t) Flow in = C(t)*Flow rate in = 20*10^6(4+sin(2t))
and C(t) Flow out = (C(t)*Flow rate out)/Volume = (20*10^6(4+sin(2t)))/150*10^9

so the dQ(t)/dt = C(t) Flow in - C(t) Flow out = [20*10^6(4+sin(2t))]-[(20*10^6(4+sin(2t)))/150*10^9]

also thanks for the pointers i think i have used them correctly

 

[Cubist]

The flow in is correct, well done! But I think your flow out is wrong. I think it would be:-

flow out = (concentration of chemicals in the lake) * (flow rate out of lake)
= (grams of toxic chemicals in the lake) / (volume of lake) * (flow rate out of lake)

This uses Q(t) not C(t)