Salt mixture A tank initially contains 100 gal of brine in which 50 lb of salt are dissolved. A

step 1 It's Clarissa enumerate here. So for part A, we know that a brine containing two pounds per gall

Salt mixture A tank initially contains 100 gal of brine in...

Salt mixture A tank initially contains 100 gal of brine in which 50$\mathrm { lb }$ of salt are dissolved. A brine containing 2$\mathrm { lb } / \mathrm { gal }$ of salt runs into the tank at the rate of 5$\mathrm { gal } / \mathrm { min. }$ 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 d. Write down and solve the initial value problem describing the e. Find the concentration of salt in the tank 25 min after the process starts.

Salt mixture A tank initially contains 100 gal of brine in which 50$\mathrm { lb }$ of salt are dissolved. A brine containing 2$\mathrm { lb } / \mathrm { gal }$ of salt runs into the tank at the rate of 5$\mathrm { gal } / \mathrm { min. }$ 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
d. Write down and solve the initial value problem describing the
e. Find the concentration of salt in the tank 25 min after the process starts.

Key Concepts

Initial Value Problems

Initial value problems involve solving differential equations with specified initial conditions. In the context of mixing problems, the initial amount of solute or volume in the tank acts as the starting point from which the future behavior of the system is determined, allowing for the complete solution of the model.

First Order Linear Differential Equations

First order linear differential equations are used to model systems where the rate of change of a quantity is proportional to the current value plus an added function. In mixing problems, such equations capture the dynamics of solute concentration affected by both the inflow of a substance at a constant rate and the proportionate removal based on the current concentration.

Mixing Problems

Mixing problems involve determining the concentration of a substance within a container where a solution with a known concentration is added while a mixture is simultaneously drained. These problems frequently lead to differential equations because the amount of the substance in the tank changes over time due to the inflow and outflow processes.

Rate of Change

The concept of rate of change is used to describe how a quantity, such as the amount of salt, changes over time. In mixing problems, the rates at which the solute enters and leaves the tank are key factors in setting up the differential equations that model the system’s behavior.

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