Salt Concentration Over Time

A tank contains 1000 liters of brine with 15 kilograms of dissolved salt. Pure water is entering the tank at a rate of 10 liters per minute, and the tank drains at the same rate. Determine how much salt is in the tank after 'T' minutes.

In this problem, we explore the concept of a mixing problem through a classic differential equations scenario involving a tank of brine solution. The core focus is on setting up and solving a first-order linear differential equation that models the change in salt concentration over time. This type of problem is often used to illustrate how systems can reach equilibrium under a steady flow and, more broadly, introduce concepts of rate of change in contexts other than simple motion.

The main strategy for solving the problem involves recognizing the balance between the rate at which pure water enters the system, diluting the brine, and the rate at which the mixed solution exits the tank, maintaining steady-state conditions. The challenge is formulating the appropriate differential equation that represents this dynamic system. Understanding the principle of conservation of mass and how it applies to the system's flow conditions is critical. By setting up the problem correctly, students can gain valuable insights into the transient behavior of solutions, which are common themes in engineering and physical sciences.

This problem is an application of first-order linear ordinary differential equations, a fundamental tool in mathematical modeling. In practice, such modeling techniques extend far beyond simple mixing and are vital in fields like chemistry, environmental science, and engineering, where predicting the behavior of systems over time is essential.