Suppose that a large mixing tank initially holds 300 liters of water in which 50 kg of salt have been dissolved. Another brine solution, with a concentration of 250 g/L, is pumped into the tank at a rate of 5 L/min, and the tank is also being emptied at a rate of 5 L/min. The fluid in the tank is being constantly stirred, and we can assume that, at any moment, the concentration is uniform throughout the tank. If A(t) represents the amount of salt (in kg) in the tank at time t, find a first-order differential equation for A(t), and write down the initial condition that goes with this equation.

Explanation:

Thus, we find

(input rate of brine)

Since the solution is pumped out at a slower rate, thus it is accumulating

at the rate of

Thus, after t minutes there will be gallons in tank

(concentration of salt in outflow) \cdot (output rate of brine)

Now, we substitute by these results in the to get

resonant frequency

we have been given a quadratic function and we need to restrict the domain such that it becomes a one to one function.

we know that vertex of this quadratic function occurs at (5,2).

further, we know that range of this function is .

if we restrict the domain of this function to either or , it will become one to one function.

let us know find its inverse.

upon interchanging x and y, we get:

let us now solve this function for y.

hence, the inverse function would be if we restrict the domain of original function to and the inverse function would be if we restrict the domain to .