Salt mixing tank differential equation
A tank initially contains 5 kg of salt dissolved in 3000 litres of water. Salty water that contains 0.1 kg of salt per litre of water enters the tank at a rate of 20 litres per minute. The solution is kept thoroughly mixed and drains from the tank via a tap at the same rate of 20 litres per minute.
By considering concentration, explain whether the quantity of salt in the tank increases with time.
Hints, answer checking and worked solutions come with a Season Pass.
Let denote the quantity of salt, in kilograms, in the tank at time minutes.
Show that satisfies the differential equation .
Hints, answer checking and worked solutions come with a Season Pass.
Using Euler’s method with a step size of 15 minutes, find , the approximate quantity of salt in the tank after 30 minutes.
Give your answer in kilograms, correct to two decimal places.
Hints, answer checking and worked solutions come with a Season Pass.
Use calculus to solve the differential equation , expressing in terms of .
Hints, answer checking and worked solutions come with a Season Pass.
What value does the quantity of salt in the tank approach as time approaches infinity?
Give your answer in kilograms.
Hints, answer checking and worked solutions come with a Season Pass.
Find the time taken for the quantity of salt in the tank to reach 100 kg.
Hints, answer checking and worked solutions come with a Season Pass.
When the quantity of salt in the tank reaches 100 kg, the tap draining the tank is turned off. Assume that the tank does not overflow and there is no change to the inflow rate.
After the tap is turned off, how many minutes does it take for the concentration of salt in the tank to reach kg L?
Hints, answer checking and worked solutions come with a Season Pass.