A tank contains 1000 L of pure water. Brine that contains...

A tank contains 1000 L of pure water. Brine that contains 0.05 kg of salt per liter of water enters the tank at a rate of 5 $\mathrm{L} /$ min. Brine that contains 0.04 kg of salt per liter of water enters the tank at a rate of 10 $\mathrm{L} /$ min. The solution is kept thoroughly mixed and drains from the tank at a rate of15 $\mathrm{L} / \mathrm{min} .$ How much salt is in the tank (a) after $t$ minutes and (b) after one hour?

A tank contains 1000 L of pure water. Brine that contains
0.05 kg of salt per liter of water enters the tank at a rate of
5 $\mathrm{L} /$ min. Brine that contains 0.04 kg of salt per liter of water
enters the tank at a rate of 10 $\mathrm{L} /$ min. The solution is kept thoroughly mixed and drains from the tank at a rate of15 $\mathrm{L} / \mathrm{min} .$ How much salt is in the tank (a) after $t$ minutes and (b) after one hour?

Key Concepts

Constant Volume Systems

A constant volume system is one in which the total volume of the solution remains unchanged over time, typically because the inflow rate equals the outflow rate. This characteristic simplifies the analysis of the concentration of substances in the tank, since changes in the amount of the substance are not complicated by changes in volume.

Rate of Change

The concept of rate of change is fundamental in understanding how the amount of a substance changes over time due to various inputs and outputs. It quantifies the net effect of the inflow and outflow of the substance, leading to the formulation of the balance equation that describes the dynamics of the system.

Initial Conditions

Initial conditions refer to the known state of the system at the start of the process, critical for determining the particular solution to a differential equation. In mixing problems, the initial amount of the substance in the tank is used to find the constant of integration in the solution of the differential equation.

Differential Equations

Differential equations are used to model the rate at which a quantity changes over time. In the context of mixing problems, a first?order linear differential equation is formulated by equating the rate of salt coming into the tank minus the rate of salt leaving the tank to the time derivative of the salt amount.

Mixing Problems

Mixing problems involve analyzing the behavior of a substance (such as salt) within a solution in a vessel where fluid flows in and out. These problems require understanding how the inflow of a substance at a certain concentration and the outflow of the solution affect the overall amount of the substance in the tank over time.

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