# Brine-ish Question with Linear Equations

• justine411
In summary: You can see that if you differentiate both sides with respect to t, you get: dX/dt= 10X/(400-6t) Which is the same as the equation you started with.
justine411

## Homework Statement

we have a tank with 400L of water with Cl, 0.02kg of Cl. Fresh water is pumped in at 4L/s and out at 10L/s. Find the amount of Cl in the tank as a function of t.

## Homework Equations

Use of differential equations

## The Attempt at a Solution

I set up V(t)=400-6t in Liters and seconds. then I want
dy/dx=(rate in)-(rate out) for Cl flowing. But no Cl is coming in correct? So rate in ends up being zero, which confuses me because then everything ends up equalling zero from what I've done.

If you want to see more of what I've done, please post. Am I wrong to say rate in is 0?

rate in= (4L/s)(0g/L)=0
rate out=(10L/s)(y(t)/(1000-6L))
then dy/dt-(10y)/(1000-6t)=0
I(t)=(1000-6t)^-10/6 (took a few steps to find, but I think it's right).

sorry...it seems like a lot of people have looked at my problem...what is unclear about it? Why can't anyone help? Is there something I could do?

I end up with integral ((400-6t)(y))prime=integral 0dt, and that can't be right.
PLEASE I AM DYING TO KNOW WHAT I'M DOING WRONG!

justine411 said:

## Homework Statement

we have a tank with 400L of water with Cl, 0.02kg of Cl. Fresh water is pumped in at 4L/s and out at 10L/s. Find the amount of Cl in the tank as a function of t.

## Homework Equations

Use of differential equations

## The Attempt at a Solution

I set up V(t)=400-6t in Liters and seconds. then I want
dy/dx=(rate in)-(rate out) for Cl flowing. But no Cl is coming in correct? So rate in ends up being zero, which confuses me because then everything ends up equalling zero from what I've done.

If you want to see more of what I've done, please post. Am I wrong to say rate in is 0?

Why would "everything end up being 0"? Yes, there is no Cl flowing in, but there certainly is Cl flowing out. Let X(t) be the amount of Cl in the tank at time t seconds. The volume of water at that time is, as you say, 400-6t so the concentration of Cl is X/(400-6t) kg/liter. Since the water is flowing out at 10 l/min, Cl is flowing out at 10X/(400-6t) kg/min.
dX/dt= -10X/(400-6t). Looks like a simple separable equation to me.

By the way, don't get upset if no one responds within a few minutes- some of us have lives to live!

Thanks! I wondered if it was a separable equation, but because the rate in is different from the rate out, isn't it linear, but not separable? That was sort of also holding me back...

Actually, it is both linear and separable.

## 1. What is "Brine-ish Question with Linear Equations"?

"Brine-ish Question with Linear Equations" is a scientific term that refers to a specific type of problem involving brine solutions and linear equations. It is commonly used in chemistry and physics to describe a situation where the concentration of salt in a brine solution changes over time due to various factors.

## 2. How do linear equations relate to brine solutions?

Linear equations are used to mathematically model the concentration of salt in brine solutions. The concentration of salt can be represented as a variable in the equation, and the other variables can represent factors such as time and temperature that affect the concentration. By solving the equations, scientists can predict how the concentration of salt will change over time.

## 3. What are some real-life applications of "Brine-ish Question with Linear Equations"?

There are many real-life applications of "Brine-ish Question with Linear Equations". For example, it can be used to study the salinity of oceans and how it affects marine life, the salinity of soil and how it affects plant growth, and the salinity of water sources and how it affects drinking water quality.

## 4. What are the key factors that influence the concentration of salt in brine solutions?

The key factors that influence the concentration of salt in brine solutions include the initial concentration of salt, the rate at which the solution is stirred, the temperature, and the amount of time that has passed since the solution was created. Other factors, such as the addition of more salt or the evaporation of water, can also play a role.

## 5. What are some potential challenges when solving "Brine-ish Question with Linear Equations"?

One potential challenge when solving "Brine-ish Question with Linear Equations" is the accuracy of the initial data. Small errors in measurements or uncertainty about the initial conditions can lead to significant differences in the predicted concentration of salt. Additionally, the equations used may not take into account all the variables that could affect the concentration of salt, leading to potentially inaccurate predictions.

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