Mixing and Circuit Problems

Rate in minus rate out, written as an equation.

12 problems · about 24 minutes · A-CED.A.1, A-CED.A.2

What this lesson teaches

The student sets up and interprets linear models for mixing tanks and simple circuits.

For anything that accumulates, the rate of change is the rate in minus the rate out.
The outflow depends on the current concentration, which makes the equation linear.
Setting the rate to zero gives the steady state the system approaches.

Straightforward practice. Get the method working first.

5 problems

dS/dt = 8 − S/30. What is the steady-state amount?
Answer 240
Why 240.
How is a mixing problem set up?
Answer The rate of change equals the rate in minus the rate out.
Why In minus out.
2 grams per litre entering at 10 litres per minute. What is the rate in?
Answer 20
Why Multiply.
dS/dt = 20 − S/50. What is the steady-state amount?
Answer 1000
Why 20 times 50.
A 300 litre tank holding 900 grams of salt. What is the concentration in grams per litre?
Answer 3
Why 900 divided by 300.

The same ideas with more to keep track of.

3 problems

Inflow 4 litres per minute and outflow 4 litres per minute. Does the volume change? 1 yes, 0 no.
Answer 0
Why They balance exactly.
dS/dt = 12 − S/25. What is the steady-state amount?
Answer 300
Why 12 times 25.
At the steady state, what is dS/dt?
Answer 0
Why Nothing is accumulating.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each part of the model to what it represents.
Answer Inflow rate times incoming concentration → Rate in; Outflow rate times current concentration → Rate out; The difference between them → The rate of change of the amount
Why Accumulation is in minus out.
A 100 litre tank of pure water, so no salt at all. What is the initial concentration?
Answer 0
Why No salt present.
The Inflow: Brine at 4 grams per litre entering at 6 litres per minute. What is the rate in, in grams per minute?
Answer 24
Why 24.
The Steady State: dS/dt = 15 − S/40. What is the steady-state amount of salt, in grams?
Answer 600
Why 600 grams.