Differential Equation - Brine Solution Entering Tank

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Homework Help Overview

The problem involves a tank initially containing pure water into which a brine solution is introduced. The task is to determine the amount of salt in the tank over time and the specific time when the concentration of salt in the outflow reaches a certain level. The subject area is differential equations, particularly focusing on modeling and solving equations related to rates of change in a mixing scenario.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the formulation of the differential equation and its solution. There is a focus on clarifying the conditions for part (b) of the problem, with some questioning the initial assumption regarding the amount of salt in the tank.

Discussion Status

The discussion is ongoing, with participants providing insights into the modeling of the problem and emphasizing the importance of showing work for effective assistance. There is a recognition of the need to clarify the interpretation of the problem's requirements, particularly regarding the time when the salt concentration reaches 1 lb/gal.

Contextual Notes

Participants note the importance of adhering to forum guidelines by presenting their work for review. There is also mention of specific numerical expectations related to the timing of the concentration reaching the desired level.

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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.
 
Last edited:
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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!
 
Last edited:
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!
 
Last edited:

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