Need some help with Calculus differential equation applications(video explanation would help)

Question

A tank contains 8,000 L of brine with 16 kg of dissolved salt. Pure water enters the tank at a rate of 80 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate.

(a) How much salt is in the tank after t minutes?

(b) How much salt is in the tank after 30 minutes? (Round your answer to one decimal place.)

Answer 1

16/8000=.002kg/L

With no more salt entering the tank; it's losing .002kg/L*(80)=.16kg/min.

Answer 2

Hello Angie,

I will go over how to set up the differential equation. Then you will have to solve it to get the answer to

part a and substitute t = 30 to get the answer to part b.

First, you need to translate this situation into a differential equation. The function variable y will represent kilograms of salt.

So let's say y(t) = kg salt after t minutes.

Then the differential equation will have dy/dt = ...

What are the units for dy/dt? It would have y units over t units so it is kg/min.

We start with 16 kg of salt so y(0) = 16.

The differential equation will be dy/dt = rate in - rate out

The amount of brine will stay at 8000 L since the same amount of liquid goes in and out.

The water comes in at 80 L/min and it has no salt.

Remember that the diff. eq. must be kg/min not L/min. How do we get that for the rate-in part?

(80 L/min)(0 kg salt/L) = 0 kg salt/min so the rate-in is 0.

dy/dt = 0 - rate out

The rate out is the mixture which will have less salt in it as time goes by because of the pure water coming in while the salted brine goes out. We know the rate is 80 L/min again, but how do we represent the

kg salt/L that is going out? This is the hardest part. We have to use the variable y since the amount of salt in the tank changes over time.

It would be y/8000 kg/L since y tells the kg of salt after t minutes.

This gives us:

dy/dt = 0 - (80 kg/min)(y/800 kg/L) so the differential equation is dy/dt = - y/10 since 80/800 is 1/10.

This is a separable differential equation so use the correct method to solve it and finish the problem.

You can always hire a tutor if you need more help on this topic or others.