Optimistic Outlook: Solving Nitric Acid Concentration in a Tank

How can we determine the volume of nitric acid in the tank after t minutes?

Given the data that a nitric acid solution flows into a large tank at a constant rate, how can we calculate the volume of nitric acid in the tank over time?

Answer:

The volume of nitric acid in the tank after t minutes can be determined by the equation x(t) = −39e^(-0.03t) + 40.

In order to calculate the volume of nitric acid in the tank after t minutes, we need to consider the input and output rate of the solution. The concentration of nitric acid inside the tank is measured in liters and changes over time due to the inflow and outflow of the solution.

By modeling the problem with the input rate I(t) and output rate O(t), we can derive the differential equation dx/dt + 0.03x = 1.2. This linear equation represents the change in volume of nitric acid in the tank over time.

The initial condition x(0) = 1 is used to solve for the constant C in the equation, which results in x(t) = −39e^(-0.03t) + 40. This formula allows us to calculate the volume of nitric acid in the tank at any given time t.

Therefore, by applying the given data and solving the differential equation, we can determine the volume of nitric acid in the tank after t minutes and track its concentration over time with an optimistic outlook.