Final answer:
An initial value problem for the quantity of salt in a tank at time t can be set up by establishing the initial salt quantity and then creating a differential equation to represent the rate of change of salt in the tank over time.
Step-by-step explanation:
To set up an initial value problem for the quantity of salt in a tank at time t minutes, we need to establish the initial conditions and rate of change of the salt quantity. Typically, such a problem can be expressed in the form of a differential equation and an initial condition. If Q(t) represents the quantity of salt in the tank at time t, and we know that initially at time t=0, the amount of salt is Q(0), then our initial condition is Q(0)= some known value in kilograms.
The rate of change of Q(t) can depend on various factors such as the rate at which the water (carrying salt) is added to the tank, the concentration of the salt in the incoming water, the volume of the tank, and the rate at which the mixture is leaving the tank. Assuming we are given the rate at which salt is arriving (in kg/min) as r(t) and the rate at which the salt mixture leaves the tank, we can set up a differential equation dQ/dt = r(t) - outflow rate in terms of Q. Finally, we can combine this equation with the initial condition to form the complete initial value problem.