Answer:
To construct a mathematical model for the number of pounds of salt in tanks A, B, and C at time t, we need to consider the rate of change of salt in each tank. Let's denote the number of pounds of salt in tank A as x1(t), in tank B as x2(t), and in tank C as x3(t).
From the given information, we can determine the rate at which salt is entering or leaving each tank. The rate of salt entering tank A is 8 gallons per minute, the rate of salt entering tank B is 6 gallons per minute, and the rate of salt entering tank C is 5 gallons per minute.
We also know that there are mixtures being added to each tank. The mixture entering tank A has a flow rate of 3 gallons per minute, the mixture entering tank B has a flow rate of 1 gallon per minute, and there is no mixture being added to tank C.
Based on these rates, we can set up the following differential equations:
d(x1)/dt = 8 - 3
d(x2)/dt = 6 - 1
d(x3)/dt = 5
These equations represent the rate of change of salt in each tank with respect to time. The right-hand side of each equation represents the net inflow rate minus the outflow rate.
To solve these differential equations, we need initial conditions for each tank. Let's assume that initially there are no pounds of salt in any of the tanks. Therefore, x1(0) = x2(0) = x3(0) = 0.
We can now solve these differential equations using standard techniques such as separation of variables or numerical methods like Euler's method or Runge-Kutta methods.
Once we have solved these equations, we will have mathematical models for the number of pounds of salt in tanks A, B, and C at any given time t.
Step-by-step explanation: