Suppose that a tank initially contains 80 liters of pure water. At a given instant (taken to be t = 0) a salt solution containing 0.5 kg of salt per liters flows into the tank at the rate of 8 liters min^-1. The liquid in the tank is kept homogeneous by constant stirring. Also, at time t = 0 liquid is allowed to flow out from the tank at the rate of 12 liters min^-1. Show that the amount of salt x(t) (in kg) in the tank at time t (min) is determined by

Figure (7.6)

Solution:

The situation is illustrated in Figure 7.6. Since x(t) denotes the amount of salt in the tank at time , the rate of increase of the amount of salt in the tank is dx/dt, and is given by

rate of inflow of salt — rate of outflow of salt (7.4)

The rate of inflow of salt is kg litre litres min kg min

The rate of outflow of salt is c(rate of outflow of liquid) = c12 litres min

=12c (in kg min)

where c(t) is the concentration of salt in the tank (in kg litre). The concentration at time t is given by

c(t) = (amount of salt in the tank at time t )/(volume of liquid in the tank at time t)

After time t (in min) 8t litres have entered the tank and 12t litres have left. Also, at t = 0, there were 80 litres in the tank. Therefore the volume V of liquid in the tank at time t is given by

(Note that only if min; after this time the liquid will flow out as quickly as it flows in and none will accumulate in the tank.) Thus, the concentration c(t) is given by

so that

rate of outflow of salt

or

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