# The Ordinary Differential Equations Project

## Section6.5Projects for Laplace Transforms

### Project6.5.1.Project—Tank Interrupt Mixing.

Water containing 2 kg of salt per liter (L) is poured into a tank at a rate of 1 L/min, and the well-stirred mixture leaves at the same rate. After 10 min, the process is stopped and fresh water is poured into the tank at a rate of 1 L/min, with the new mixture leaving at 1 L/min. We seek to model this situation in order to determine the amount (kg) of salt in the tank at the end of 20 minutes if there were 100 L of pure water initially in the tank.
1
This project is adapted from Norman Loney (2018), “7-040-S-TankInterruptMixing,” https://www.simiode.org/resources/5103.

#### (a)

Build a mathematical model for the amount of salt in kg in the tank at time $$t$$ in min with two parts, first over the time interval $$[0, 10]$$ minutes and then over the time interval $$[10, 20]$$ minutes. You can assume instantaneous mixing of the two different solutions.

#### (b)

Build a mathematical model for the amount of salt in kg in the tank at time $$t$$ in min using the Heaviside or Unit Step functions to represent the sudden change in input water types and apply Laplace Transforms and Inverse Laplace Transforms to determine the amount of salt in the tank over the entire interval $$[0, 20]$$ minutes. Then compare your results with Activity 1.