A contains 200 liters of solution in which is dissolved 40 kg of
salt. Tank B contains 400 liters of solution in which are dissolved
80 kg of salt. Pure water flows into tank A at rate of 10 liters per
second. There is a drain at the bottom of tank A. Solution leaves
tank A via this drain at a rate of 10 liters per second and flows
immediately into tank B at the same rate. A drain at the bottom of
tank B allows the solution to leave tank B, also at a rate of 10
liters per second. What is the salt content in the tank B after 1
minute?
x(t) :amount of salt in tank A
dx/dt: rate of salt changing with respect to time in tank A
y(t) : amount of salt in tank B
dy/dt: rate of salt changing with respect to time in tank B
rate of change of substance = rate in of substance
rate in/out =flow rate(or volume rate) in/out x concentration within the fluid entering/exiting
concentration = amount of salt/volume of solution
see solution 3 http://www.math.cmu.edu/~doffner/teaching/122/quiz7sol.pdf
To find the salt content in tank B after 1 minute, we need to find the rate at which salt is entering and leaving tank B.
Let's first find the rate at which salt is entering tank B from tank A. The flow rate of solution from tank A to tank B is 10 liters per second, and the concentration of salt in tank A is 40 kg / 200 liters = 0.2 kg/liter. So, the rate of salt entering tank B from tank A is:
Rate of salt entering tank B from tank A = Flow rate x Concentration
= 10 liters/second x 0.2 kg/liter
= 2 kg/second
Now, let's find the rate at which salt is leaving tank B through the drain. The flow rate of solution leaving tank B is also 10 liters per second, and the concentration of salt in tank B is 80 kg / 400 liters = 0.2 kg/liter. So, the rate of salt leaving tank B is:
Rate of salt leaving tank B = Flow rate x Concentration
= 10 liters/second x 0.2 kg/liter
= 2 kg/second
Since the rate of salt entering tank B from tank A is equal to the rate of salt leaving tank B, the salt content in tank B remains constant at 80 kg throughout the process.
Therefore, after 1 minute, the salt content in tank B will still be 80 kg.