Answer: (a) Let's call the amount of water in the tank after t minutes "W(t)". We know that the rate of change of water in the tank is equal to the rate of brine being pumped in minus the rate of solution being pumped out. Therefore, we can write:
dW(t)/dt = 9 - 5 = 4
This is a first-order linear differential equation, and it can be solved to find W(t). Integrating both sides with respect to t, we get:
W(t) = 4t + C, where C is a constant of integration.
To find the value of C, we can use the initial condition that the tank initially contains 50 litres of water, so:
W(0) = 50
Plugging in t = 0, we get:
50 = 4 * 0 + C
Therefore, C = 50.
So the expression for the amount of water in the tank after t minutes is:
W(t) = 4t + 50
(b) Let's call the amount of salt in the tank after t minutes "x(t)". We know that the rate of change of salt in the tank is equal to the rate of brine being pumped in times the concentration of salt in the brine minus the rate of solution being pumped out times the concentration of salt in the solution. Therefore, we can write:
dx(t)/dt = 11 * 9 - x(t) * 5
This is a first-order linear differential equation, and it describes the amount of salt in the tank after t minutes.
Explanation: