A tank initially contains 100 gal of brine in which 50 lb of...

A tank initially contains 100 gal of brine in which 50 lb of salt is dissolved. A brine containing 2 Ib/gal of salt runs into the tank at the rate of 5 gal/min. The mixture is kept uniform by stirring and flows out of the tank at the rate of 4 gal/min. a. At what rate (pounds per minute) does salt enter the tank at time $t ?$ b. What is the volume of brine in the tank at time $t ?$ c. At what rate (pounds per minute) does salt leave the tank at time $t ?$ d. Write down and solve the initial value problem describing the mixing process. e. Find the concentration of salt in the tank 25 min after the process starts.

A tank initially contains 100 gal of brine in which
50 lb of salt is dissolved. A brine containing 2 Ib/gal of salt runs into the tank at the rate of 5 gal/min. The mixture is kept uniform by stirring and flows out of the tank at the rate of 4 gal/min.
a. At what rate (pounds per minute) does salt enter the tank at time $t ?$
b. What is the volume of brine in the tank at time $t ?$
c. At what rate (pounds per minute) does salt leave the tank at time $t ?$
d. Write down and solve the initial value problem describing the mixing process.
e. Find the concentration of salt in the tank 25 min after the process starts.

Key Concepts

Concentration in Mixtures

Concentration is defined as the amount of solute per unit volume of the solution. In the context of mixing problems, calculating the concentration involves tracking both the total amount of solute (like salt) and the dynamic volume of the mixture, allowing one to understand the system’s behavior over time.

Volume Dynamics

In many real-world mixing problems, the volume of the mixture is not constant because the rates of inflow and outflow may differ. Understanding how the volume changes over time is crucial since it affects the concentration of the solute in the mixture.

Initial Value Problems (IVP)

An initial value problem provides both the differential equation describing a system’s dynamics and the initial condition of the system. This combination ensures a unique solution that aligns with the system’s starting state, making it essential for accurately modeling processes in time.

Differential Equations

Differential equations express the relationship between a function and its derivatives, capturing how a physical quantity changes over time. In mixing problems, they are used to model the evolution of salt concentration, often resulting in linear first-order equations that can be solved using standard techniques.

Rate of Change Models

This concept involves setting up the system's behavior by equating the rate of change of a quantity (such as salt) to the difference between the input rate and the output rate. It underpins the construction of models for dynamic systems where an entity is being added and removed simultaneously.

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