problem with composing an equation

[Nickik]

Hello,
I have problem with composing an equation to this:

The tank holds 100 gallons of brine in which 40 pounds of salt have been dissolved. It is desirable to reduce the salt concentration to 0.1 pounds per gallon by pouring clean water at 5 gallons per minute and letting the mixture flow (which is kept evenly by mixing) at the same speed. How long will it take?

I calculated it would take 29 minutes but i dont know how to compose an equation for this.

Thank you in advance for any help.

 

[Deleted member 4993]

Hello,
I have problem with composing an equation to this:

The tank holds 100 gallons of brine in which 40 pounds of salt have been dissolved. It is desirable to reduce the salt concentration to 0.1 pounds per gallon by pouring clean water at 5 gallons per minute and letting the mixture flow (which is kept evenly by mixing) at the same speed. How long will it take?

I calculated it would take 29 minutes but i dont know how to compose an equation for this.

Thank you in advance for any help.

You said:

I calculated it would take 29 minutes​

How did you calculate that? We can help you formulate the equation through your "work".

 

[JeffM]

Are you studying differential equations or algebra?

 

[Dr.Peterson]

Hello,
I have problem with composing an equation to this:

The tank holds 100 gallons of brine in which 40 pounds of salt have been dissolved. It is desirable to reduce the salt concentration to 0.1 pounds per gallon by pouring clean water at 5 gallons per minute and letting the mixture flow (which is kept evenly by mixing) at the same speed. How long will it take?

I calculated it would take 29 minutes but i dont know how to compose an equation for this.

Thank you in advance for any help.

Suppose the tank holds x pound of salt at a given time t. How fast is x decreasing due to the mixture flowing out? (Or, if you prefer, x could be the current concentration of salt.)

If there's something about the problem that you don't understand, please ask specific questions, or tell us your current thinking, so we have something to answer.

 

[Eugenio]

Hello,
I have problem with composing an equation to this:

The tank holds 100 gallons of brine in which 40 pounds of salt have been dissolved. It is desirable to reduce the salt concentration to 0.1 pounds per gallon by pouring clean water at 5 gallons per minute and letting the mixture flow (which is kept evenly by mixing) at the same speed. How long will it take?

I calculated it would take 29 minutes but i dont know how to compose an equation for this.

Thank you in advance for any help.

You need to do a mass balance.

For the overal, flow in is equal to flow out:

[MATH]F_{in} = F_{out}=F[/MATH]
This means that the volume V remains constant and the concentration at time t is [MATH]c(t)=\frac{m(t)}{V}[/MATH]
where [MATH]m(t)[/MATH] is the mass of salt at time t. You can write its balance like this

[MATH]\frac{dm}{dt}=m_{in}-m_{out}=0-c(t)*F=-\frac{m(t)*F}{V}[/MATH]
The solution for this equation is [MATH]m(t)=m(0)*e^{-\frac{F*t}{V}}[/MATH]
And in terms of the concentration, and replacing the data of the problem, the concentration c at time t is

[MATH]c(t)=0.4e^{-0.05t}[/MATH]
So, when [MATH] c = 0.1\Rightarrow t = 27.7[/MATH] minutes