# The Mixing Problem: Understanding the Dilution of Brine in a Tank

• ank91901
In summary, the conversation is about solving a differential equation problem involving a tank initially holding 100 gallons of brine with 30 pounds of dissolved salt. Fresh water enters the tank at a rate of 3 gallons per minute, while brine flows out at the same rate. The problem is to find the salt content of the brine after 10 minutes and when it will be 15 pounds. The conversation then moves on to a second problem where the fresh water rate is changed to 2 gallons per minute. The solution for the first problem is given, but the person is struggling to solve the second problem and understand how the change in fresh water rate affects the solution inside the tank. The expert summarizer cannot provide a solution without more
ank91901
Hi all. I'm trying to help a friend work a differential equation problem since I've already taken this course and gotten an A in it, but this other problem has me a little confused.

The book is ODE's by Tenenbaum and Pollard, chapter 3 lesson 15a number 2.

#1) Tank initially holds 100 gal of brine containing 30 lb of dissolved salt. Fresh water flows into the tank at a rate of 3 gal/min and brine flows out at the same rate. (a) Find the salt content of the brine at the end of 10 minutes and (b) When will the salt content be 15 lb?

This one I've solved easily but number 2 says,

#2) Solve problem 1 if 2gal/min of fresh water enter the tank instead of 3 gal/min.

The solution on the next page says it should be (a) x=30(1-.01t)^3 which, when 10 is plugged for t gives 21.87 lb.

Can someone explain what I'm missing? Shouldn't the rate in still be zero?

It is very difficult to explain what you are missing when you haven't explained what you did to try to solve the problem. Can you tell us what your solution is for both parts?

And it is unclear what you mean when you say "the rate in should still be zero". The rate of water flowing into the tank? This is clearly not zero.

Well for #1 I got a) x=30e^(-.03t), x(10)= 22.2 lb. b) 23.1 minutes

For number I've tried finding the integrating factor but that still leaves me with the question of whether the rate of fresh water in changing from 3gal/min to just 2gal/min has anything to do with the amount of solution in the tank. When I use rate in= 2gal/min and try the integrating factor method I get a solution that looks like this: x=-2/.03+(30+2/.03)e^(-.03t) which is a ridiculous result. Any clues?

And the rate of solution coming in is zero because the problem says it is fresh water flowing in. So rate of solution flowing in is zero but fresh water flowing in is 2 gal/min in problem 2 and 3 gal/min in problem 1.

Last edited:
I guess my real question isn't "can someone help me solve this?" It's more "can someone explain how the rate of fresh water coming in affects the solution inside the tank?"

The fresh water flowing into the tank dilutes the brine solution in the tank. Brine is flowing out at the same rate as fresh water is flowing into the tank, so in time, what's in the tank will be just water with no salt.

## 1. What is a mixing problem?

A mixing problem is a type of problem in mathematics that involves finding the concentration or amount of a solution that results from mixing different substances or solutions together.

## 2. What are the key factors to consider in a mixing problem?

The key factors to consider in a mixing problem are the initial amounts or concentrations of the substances being mixed, the rate at which they are being mixed, and any changes in the system over time (such as evaporation or chemical reactions).

## 3. How do you set up and solve a mixing problem?

To set up and solve a mixing problem, you first need to identify the initial amounts or concentrations of the substances being mixed. Then, you will need to use a formula or equation that relates the concentrations and amounts of the substances to solve for the final concentration or amount.

## 4. What are some real-world applications of mixing problems?

Mixing problems have many real-world applications, including in the fields of chemistry, engineering, and pharmacology. They can be used to determine the appropriate ratios of ingredients in food and beverage production, calculate the concentrations of pollutants in water, and optimize drug dosages.

## 5. What are some common mistakes to avoid when solving a mixing problem?

Some common mistakes to avoid when solving a mixing problem include not properly converting units of measurement, using the wrong formula or equation, and not taking into account any changes in the system over time. It is important to carefully read and understand the problem, and double check all calculations to avoid these errors.

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