You CAN Ace Differential Equations

### 17Calculus Subjects Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

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17calculus > differential equations > mixing and chemical concentration in fluids

 Basic Equation Chemical-In Rate Chemical-Out Rate Well-Mixed Tank Practice

On this page we discuss one of the most common types of differential equations applications of chemical concentration in fluids, often called mixing or mixture problems. The idea is that we are asked to find the concentration of something (such as salt or a chemical) diluted in water at any given time. Usually we are adding a known concentration to a tank of known volume. Sometimes the tank is being drained at the same time. There are many variations to this problem but they are usually easily set up and solved using basic differential equations techniques.

Basic Equation

There is only one basic equation you need and setting it up is the tricky part. We highly recommend that you carry your units with you as you set it up. Sometimes, mistakes can be caught when the units don't work out. The equation you need is

$$\displaystyle{ \frac{dA}{dt} = ( rate~in ) - ( rate~out ) }$$

Notes
1. Since this is a rate problem, the variable of integration is time t.
2. $$A$$ is the amount or quantity of chemical that is dissolved in the solution, usually with units of weight like kg.
3. The rates (rate in and rate out) are the rates of inflow and outflow of the chemical. The units are usually weight per unit time, like kg/min.

As you work these problems, you will notice that there are two rates involved, the rate of flow of chemical and the rate of flow of the fluid. It is important that you understand the difference when you are working these problems. We will separate the flow rates like this.
chemical-in = rate of chemical coming into the tank with units weight/time, like kg/min.
fluid-in = rate of the fluid entering the tank with units volume/time, like litres/min = L/min.
[similarly for flow rates out of the tank]
Notice in the table above, the rate in and rate out are chemical-in and chemical-out rates. Rewriting that equation we have $$\displaystyle{ \frac{dA}{dt} = ( \textit{ chemical-in-rate } ) - ( \textit{ chemical-out-rate } ) }$$.

The tricky part is knowing how to set up the rates of chemical-in-rate and chemical-out-rate. The chemical-in-rate is usually the easiest, so let's look at that first.

Chemical-In Rate

Usually we are given the concentration of the fluid coming in and the rate at which it is flowing in. For example, one of the practice problems gives the rate in as 10L/min of pure water (with no chemical or salt). There is no chemical in the solution (since it is pure water), so the amount of chemical is 0kg/L. The rate of inflow of the chemical is modeled as
$$\textit{ chemical-in } = (0 kg/L) (10 L/min) = 0 kg/min$$.
Notice that we wrote our units in this equation, so that we were easily able to see that the units L canceled, leaving the units kg/min, which is what we want.

Chemical-Out Rate

Okay, so now let's look at outflow. This is usually where the variable we call $$A(t)$$, or just $$A$$, comes in. The variable $$A$$ is the amount (in weight, like kg) of the chemical. It is dependent on t, so it is changing. The concentration in the tank at any time t is just $$A$$ divided by the volume of fluid in the tank. The volume is calculated using this equation.
volume of fluid in the tank at any time = initial amount + ( rate coming in - rate going out ) times t
Notes
1. We assume the rates coming in and going out are constant. If they are not, the above equation would need to be adjusted accordingly.
2. If the fluid is coming in and going out at the same rate, the volume of fluid in the tank is constant and equal to the initial amount.

So, for example, if we have a tank that starts out with 1000L and fluid is coming in at a rate of 10L/min and going out at a rate of 12L/min, we have
volume of fluid in the tank = 1000L + (10L/min - 12L/min)t = 1000-2t L
This equation describes the amount of fluid in the tank at any time t and this is what is divided into A to get the rate of chemical out of the tank.

Study Tip

As you work these problems, it may help to set up a table containing everything you know and everything you need to find. In the first practice problem, we set one up as an example. As you study, try different formats for the table until you come up with one that makes sense for you.

Well-Mixed Tank

When we discussed the outflow of the tank above, we had to use the idea that the tank was well mixed and that the chemical was dispersed evenly throughout the tank. If we didn't have this assumption, we would have needed a term in the equation that was a function of time describing how the chemical was dispersed in the tank. This is an extremely complicated concept and one that you will probably not come across in differential equations.

Before working some practice problems, let's watch a quick video explaining these types of problems in a bit more detail. This is pretty theoretical but it will give you another perspective that may be helpful.

### Commutant - well-mixed tank [mins-secs]

video by Commutant

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

Instructions - - Solve the following problems, giving your answers in exact form. Unless otherwise stated, assume the tanks are well-stirred.

Basic Problems

A tank contains 1000L of fluid and 15kg dissolved salt. Fresh water enters at 10L/min and the tank is draining at 10L/min. How much salt is in the tank at t minutes and after 20 minutes?

Problem Statement

A tank contains 1000L of fluid and 15kg dissolved salt. Fresh water enters at 10L/min and the tank is draining at 10L/min. How much salt is in the tank at t minutes and after 20 minutes?

Solution

One way to get your head around this problem is to build a table of the information you have and what you need. Here is one way to build a table. If this doesn't help you, try another format.

 inflow outflow fluid flow rate 10 L/min 10 L/min chemical concentration 0 kg/L A/vol kg/L chemical flow rate (10 L/min)(0 kg/L) = 0 kg/min (10 L/min)(A/vol kg/L) = 10A/vol kg/min

From the table, we can now see that we need to determine the volume of the fluid.
volume = initial amount + (rate in - rate out)t = 1000L + (10 L/min - 10 L/min)t = 1000L
Adding the two expressions in the last row, we have $$\displaystyle{ \frac{dA}{dt} = (0 ~kg/min) - (A/100~ kg/min) }$$.

In this video, she shows how carrying the units along is helpful and she explains the solution clearly. However, she does something strange with the constant of integration by replacing $$e^c$$ with $$\pm c$$. This is confusing since there are actually two mistakes here. First, $$e^c$$ is always positive, so the $$\pm$$ sign is incorrect. Secondly, since $$e^c$$ can never equal $$c$$, it just doesn't make sense. What she is actually doing is representing the unknown constant $$e^c$$ with another unknown constant but she doesn't change her notation. We would probably write $$k=e^c$$ to make it clear that they are different.

### 625 solution video

video by Krista King Math

A tank with 200 gallons of brine solution contains 40 lbs of salt. A concentration of 2 lb/gal is pumped in at a rate of 4 gal/min. The concentration leaving the tank is pumped out at a rate of 4 gal/min. How much salt is in the tank after 1 hour? How much salt is in the tank after a very long time?

Problem Statement

A tank with 200 gallons of brine solution contains 40 lbs of salt. A concentration of 2 lb/gal is pumped in at a rate of 4 gal/min. The concentration leaving the tank is pumped out at a rate of 4 gal/min. How much salt is in the tank after 1 hour? How much salt is in the tank after a very long time?

Solution

### 627 solution video

video by MIP4U

A tank initially holding 80gals of fluid with 40lbs of salt has two sources of fluid being pumped in. One source is pumping pure water in at 8 gal/min and the second source is pumping in a salt solution of 0.5lbs/gal at 2 gal/min. The tank is emptying at 10gal/min. Find the amount of salt in the tank at any time t.

Problem Statement

A tank initially holding 80gals of fluid with 40lbs of salt has two sources of fluid being pumped in. One source is pumping pure water in at 8 gal/min and the second source is pumping in a salt solution of 0.5lbs/gal at 2 gal/min. The tank is emptying at 10gal/min. Find the amount of salt in the tank at any time t.

Solution

### 1361 solution video

video by Kent Mearig

A 100 gallon tank with 50 gals of pure water has a solution of 0.4lbs/gal being pumped in at a rate of 3 gals/min. The well-mixed solution leaves the tank at a rate of 2 gals/min. Find an expression for the salt concentration at any time t.

Problem Statement

A 100 gallon tank with 50 gals of pure water has a solution of 0.4lbs/gal being pumped in at a rate of 3 gals/min. The well-mixed solution leaves the tank at a rate of 2 gals/min. Find an expression for the salt concentration at any time t.

Solution

This problem is solved in two videos. He sets up the differential equation in the first video and then solves it in the second video.

### 1362 solution video

video by SchoolOfChuck

### 1362 solution video

video by SchoolOfChuck

A 1000L tank starts out with 200L of fluid containing 10g/L of dye. Pure water is poured in at 20 L/min and the tank is being drained at a rate of 15L/min. Write the equation for the amount of dye in the tank at any time t.

Problem Statement

A 1000L tank starts out with 200L of fluid containing 10g/L of dye. Pure water is poured in at 20 L/min and the tank is being drained at a rate of 15L/min. Write the equation for the amount of dye in the tank at any time t.

Solution

### 1363 solution video

video by David Lippman

A tank contains 90 kg of salt and 2000 L of water. Pure water enters the tank at the rate of 8 L/min. The solution is mixed and drains from the tank at the rate of 4 L/min.
(a) What is the amount of salt in the tank initially?
(b) Find the amount of salt in the tank after 1.5 hours.
(c) Find the concentration of salt in the solution in the tank as time approaches infinity. (Assume your tank is large enough to hold all the solution.)

Problem Statement

A tank contains 90 kg of salt and 2000 L of water. Pure water enters the tank at the rate of 8 L/min. The solution is mixed and drains from the tank at the rate of 4 L/min.
(a) What is the amount of salt in the tank initially?
(b) Find the amount of salt in the tank after 1.5 hours.
(c) Find the concentration of salt in the solution in the tank as time approaches infinity. (Assume your tank is large enough to hold all the solution.)

Solution

### 2289 solution video

video by MIP4U

A 100 gallon tank is being filled with a solution containing 100 grams per gallon of chlorine. The tank is being filled at the rate of 10 gallons per minute. Initially the tank contains 20 gallons with chlorine content of 200 grams. The well-mixed solution leaves the tank at a rate of 5 gallons per minute. Set up but do not solve the initial value problem that models this. If they fail to stop the flow in when the tank is full, what will the eventual chlorine content be?

Problem Statement

A 100 gallon tank is being filled with a solution containing 100 grams per gallon of chlorine. The tank is being filled at the rate of 10 gallons per minute. Initially the tank contains 20 gallons with chlorine content of 200 grams. The well-mixed solution leaves the tank at a rate of 5 gallons per minute. Set up but do not solve the initial value problem that models this. If they fail to stop the flow in when the tank is full, what will the eventual chlorine content be?

$$\displaystyle{ \frac{dQ}{dt} = 1000 - \frac{5Q}{20+5t} }$$; As $$t \to \infty$$ the concentration in the tank will be the same as is pouring in, i.e $$100g/gal$$ or $$10000g$$ in the entire tank.

Problem Statement

A 100 gallon tank is being filled with a solution containing 100 grams per gallon of chlorine. The tank is being filled at the rate of 10 gallons per minute. Initially the tank contains 20 gallons with chlorine content of 200 grams. The well-mixed solution leaves the tank at a rate of 5 gallons per minute. Set up but do not solve the initial value problem that models this. If they fail to stop the flow in when the tank is full, what will the eventual chlorine content be?

Solution

Flowing In: $$100g/gal$$ flowing in at a rate of $$10gal/min$$.
Flowing Out: Concentation of $$Q~g/V~gal$$ flowing out at a rate of $$5 gal/min$$.
Initial State: $$20 gal$$ with $$200g$$ of chlorine.
Volume In Tank: initial amt + (flow in - flow out)(time elapsed), so $$V= 20 gal + (10gal/min - 5/min)t = 20+5t$$

We need to set up the equation $$\displaystyle{ \frac{dQ}{dt} = rate~in - rate~out }$$.

$$\displaystyle{ rate~in = \frac{100g}{gal} \cdot \frac{10 gal}{min} = 1000 g/min }$$

$$\displaystyle{ rate~out = \frac{Q~g}{V~gal} \cdot \frac{5gal}{min} = \frac{5Q}{20+5t} }$$

$$\displaystyle{ \frac{dQ}{dt} = 1000 - \frac{5Q}{20+5t} }$$; As $$t \to \infty$$ the concentration in the tank will be the same as is pouring in, i.e $$100g/gal$$ or $$10000g$$ in the entire tank.

Intermediate Problems

Initially a tank contains 1kg of salt dissolved in 100L of water. Salty water containing 1/4kg/L at a rate of 3L/min is added to the tank and the (stirred) solution is draining from the tank at 3L/min. Determine an equation for how much salt is in the tank at any time t.

Problem Statement

Initially a tank contains 1kg of salt dissolved in 100L of water. Salty water containing 1/4kg/L at a rate of 3L/min is added to the tank and the (stirred) solution is draining from the tank at 3L/min. Determine an equation for how much salt is in the tank at any time t.

Solution

### 626 solution video

video by Dr Chris Tisdell

A tank holds 300 gallons of brine solution with 40 lbs of salt. A concentration of 2 lbs/gal is pumped in a rate of 4 gal/min. The concentration leaving the tank is pumped out at a rate of 3gal/min. How much salt is in the tank after 12min?

Problem Statement

A tank holds 300 gallons of brine solution with 40 lbs of salt. A concentration of 2 lbs/gal is pumped in a rate of 4 gal/min. The concentration leaving the tank is pumped out at a rate of 3gal/min. How much salt is in the tank after 12min?

Solution

video by MIP4U