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A tank has brine flowing into it at 10 L per minute with salt concentration 0.1 kg per L. The...

Question:

A tank has brine flowing into it at 10 L per minute with salt concentration 0.1 kg per L. The contents of the tank are kept thoroughly mixed and the contents flow out at 10 L per minute. Initially, the tank contains 20 kg of salt in 100 L of water.

Let the amount of salt in the tank at time t be S(t). Write a differential equation for S(t) with its initial condition.

Find the Amount of Salt :

Let {eq}S(t) {/eq} be the amount of salt in the tank at time t then

{eq}r_{i}={/eq} Rate of salt inside the tank

{eq}r_{o}={/eq} Rate of salt outside the tank

{eq}c_{i}={/eq} Concentration of salt inside the tank

{eq}c_{o}={/eq} Concentration of salt outside the tank

This is a mixing tank problem. This type problem can usually be transformed into an initial value problem. To find the solution of the problem we use following formula.

{eq}\displaystyle \frac{\mathrm{d} S}{\mathrm{d} t}=r_{i}c_{i}-r_{o}c_{o} {/eq}

Answer and Explanation:

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Let S(t) be the amount of salt in the tank at time t then

{eq}\text{Inflow information}\hspace{2cm} \text{Outflow information}\\ \displaystyle ...

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First-Order Linear Differential Equations

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Chapter 16 / Lesson 3
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In this lesson you'll learn how to solve a first-order linear differential equation. We first define what such an equation is, and then we give the algorithm for solving one of that form. Specific examples follow the more general description of the method.


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