# Differential Equation - Brine Solution Entering Tank

• Onett
In summary, the problem involves a tank containing 80 gallons of pure water and a brine solution with 2 lb/gal of salt entering and leaving at the same rate of 2 gal/min. The first part (a) asks for the amount of salt in the tank at any time, which can be modeled by the differential equation S=160 - 160*e^(-t/40). The second part (b) asks for the time at which the brine leaving will contain 1 lb/gal of salt, which can be found by solving the differential equation dS/dt=4-2S/80.
Onett

## Homework Statement

A tank contains 80 gallons of pure water. A brine solution with 2 lb/gal of salt enters at 2 gal/min, and the well-stirred mixture leaves at the same rate. Find (a) the amount of salt in the tank at any time and (b) the time at which the brine leaving will contain 1 lb/gal of salt.

## Homework Equations

I'm just wondering about (b) really. I know we set S=80 below to solve it, but why?

## The Attempt at a Solution

The differential equation that gives (a) is

S=160 - 160*e^(-t/40)

where S is the amount salt in the tank at any time t.

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If you correctly modeled a diff. eq for this problem and, also correctly solved it to come up with the sol

S=160 - 160*e^(-t/40), then part b)is not a problem at all. what it is asking u is that when will S(t)=1, and not 80 as you are saying!
remember S(t) is the amount of salt that the tank contains at any time.
The diff eq for this problem is

dS/dt=Ri*Ci- (S*Ro)/(Vo+(Ri-Ro)t) , where

S--- is the amount of salt in the tank,
Ri rate in
Ro rate out
Ci concentration in
Vo the initial volume

EDIT: You haven't actually showed us what u have done at all, remember one of the forums main policy is that you must first show your work, for after the people here to give you hints!

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Oh, I'm sorry about that. I'll be sure to put up my work soon. Are you sure that what it's asking though? My notes say that I should get somewhere around 28 minutes.

A tank contains 80 gallons of pure water. A brine solution with 2 lb/gal of salt enters at 2 gal/min, and the well-stirred mixture leaves at the same rate. Find (a) the amount of salt in the tank at any time and (b) the time at which the brine leaving will contain 1 lb/gal of salt.

here it is :

dS/dt=2*2- (S*2)/(80+(2-2)*t)
dS/dt=4-2S/80, just solve this diff eq, if you haven't gone like this.

and for the part b) it is just asking you at what time t=? will S(t)=1, like i said.
NOTE: Next time show your work if you want to receive any help!

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## 1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is used to model various physical phenomena and is commonly used in science and engineering.

## 2. How is a brine solution entering a tank described by a differential equation?

The process of a brine solution entering a tank can be described by a differential equation that relates the rate of change of the concentration of the solution to factors such as the flow rate, tank volume, and initial concentration of the solution.

## 3. Why is it important to study differential equations in relation to brine solution entering a tank?

Studying differential equations allows us to understand and predict the behavior of a brine solution entering a tank. This knowledge is crucial in industries such as chemical engineering, where accurate predictions of solution concentrations are necessary for efficient and safe operations.

## 4. What are some real-life applications of differential equations in relation to brine solution entering a tank?

Differential equations are used in various real-life applications, such as in the production of saltwater for desalination plants, in the treatment of wastewater, and in the production of brine solutions for industrial processes. They are also used to model biological systems, such as the diffusion of nutrients in plant roots.

## 5. How can differential equations be solved in the context of brine solution entering a tank?

There are several methods for solving differential equations, such as separation of variables, substitution, and using integrating factors. In the context of brine solution entering a tank, the appropriate method will depend on the specific equation and initial conditions. Numerical methods, such as Euler's method, can also be used to approximate solutions.

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