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Two tanks are connected to each other through two pipes. Tank A contains 200 liters of water in which 60 grams of salt is dissolved (mixture) and Tank B contains 200 liters of pure water. The mixture and the pure water is circulated between the two tank A and tank B such that outflow of mixture from tank A to tank B is 20 Liters/min and inflow of mixture from tank B to tank A is 5Liters/ min. pure water is added to tank A at 15 Liters/min through inlet pipe and mixture from tank B is removed through outlet pipe at 15 liters/min. How much time does it take to reach such that both the tanks contain equal percentage of salt dissolved in both the tanks?

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What have you tried? Can you write the equations describing how much water is in A and B and how much salt is in each? –  Ross Millikan Jul 29 '11 at 16:56
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@user13359: Though you have been in the site for two weeks this is only your second question, so perhaps you are not aware: many of us consider it very impolite to post questions as if you were handing out a homework assignment, which is what this post does. All you are doing is quoting a problem (from a book or an assignment, presumably). If you are quoting, you may want to use a quote box. But in addition, you should tell us (i) why you are looking at this problem (self-study? Asssignment? Homework?); (ii) What course/background this is for; and (iii) your thoughts and why you are stuck. –  Arturo Magidin Jul 29 '11 at 17:17
    
I apologize for not being as responsive participant in this site as expected. I am still trying to understand how this site works. I did not mean to be impolite towards any one of the users. I had been posting the question as part of my own exploration and self-study. Any of the questions posted are not related to my field of study. I have been trying to explore the questions by modifying questions that are asked in a book.I will try to explain better in my next posting what I think I did wrong before posting the problem here. Thank you everybody! –  user13359 Aug 1 '11 at 21:26
    
The method I have followed in solving the above problem is similar to the answer that is posted here by Andre Nicolas. I have come up with differential equations for flows of mixtures in tanks A & B and determine the eigen values and its corresponding eigen vector to obtain equation at any time “t”. But I recognize now that went wrong during the process of equating the two equations and obtain final equation in terms of only “t” and got the value of t=200min and when I try to verify the answer the amount of salt quantity in the tanks as “zero” which is incorrect. THANKS FOR THE GUIDANCE. –  user13359 Aug 2 '11 at 21:11

1 Answer 1

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We model the situation by a system of two homogeneous linear differential equations with constant coefficients.

Let $x=x(t)$ be the quantity, in grams, of salt in tank A at time $t$, and $y=y(t)$ the quantity of salt in tank B at time $t$.

We are given the initial conditions $x(0)=60$ and $y(0)=0$.

We now examine the flow of salt in and out of tank A. Brine is flowing out to B at the rate of $20$ liters per minute. At any time $t$, the amount of salt in a liter of water is $x/200$, so salt is flowing out at the rate $(20/200)x$. Also, salty (after a while) water is flowing in from B at rate $5$ liters per minute. Thus salt is flowing into A at the rate $(5/200)y$. The above out-in analysis for the salt can be written as the differential equation $$\frac{dx}{dt}=-\frac{20}{200}x +\frac{5}{200}y \qquad\qquad\text{(Equation $1$)}$$

Now we do a similar analysis for tank B. Salt is flowing in from A at the rate of $(20/100)x$. Salt is flowing out to A at the rate $(5/200)y$, and to the outlet pipe at the rate $(15/200)y$, for a combined out rate of $(20/200)y$. This yields the differential equation $$\frac{dy}{dt}=\frac{20}{200}x -\frac{20}{200}y \qquad\qquad\text{(Equation $2$)}$$

We run through the solution process. Let $M$ be the matrix $$\begin{pmatrix} -\frac{20}{200}& \frac{5}{200}\\ \frac{20}{200}& -\frac{20}{200} \end{pmatrix}$$ We first find the eigenvalues and associated eigenvectors of the matrix $M$. Standard calculation shows that the eigenvalues are $-1/20$ and $-3/20$. The vector $(1,2)$ (written as a column vector) is an eigenvector for eigenvalue $-1/20$; the vector $(1,-2)$, again written as a column vector, is an eigenvector for eigenvalue $-3/20$.

Theory tells us that the general solution $(x,y)$ is given by $$(x,y)=Ce^{-t/20}(1,2)+De^{-3t/20}(1,-2)$$ where $C$ and $D$ are constants. Less compactly, $$x=Ce^{-t/20}+De^{-3t/20} \qquad \text{and}\qquad y=2Ce^{-t/20}-2De^{-3t/20}.$$

From the fact that $x(0)=60$ we get $C+D=60$. From $y(0)=0$ we get $2C-2D=0$. So $C=D=30$, and therefore $$x=30e^{-t/20}+30e^{-3t/20} \qquad \text{and}\qquad y=60e^{-t/20}-60e^{-3t/20}.$$

Now we know everything, so we can answer any question. We were asked when there are equal percentages of salt in both tanks. The tanks are of equal size, so equal percentage happens when the amounts of salt are equal, that is, at the time $t$ when $x=y$. Thus $$30e^{-t/20}+30e^{-3t/20}=60e^{-t/20}-60e^{-3t/20}.$$
This simplifies to $3e^{-3t/20}=e^{-t/20}$, then to $e^{t/10}=3$. Take the natural logarithm of both sides: $t=10\ln 3\approx 10.98$.

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