Math Is Fun Forum

b31den

Guest

Differential Equations - Brine Problem

In general I have no problems with the actual math, I'm having trouble setting up the equation for this problem.

Water flows thrugh a purifying machine at the rate of 30 gals/min. The machine removes 10% of the impurities as the water flows through. Suppose I have 1000 gals of water containing 5 gals of impurities. How long do I have to run it through the machine to reduce the amount of impurity to 2 quarts (a half-gallon)?

i know s'(t) = input - output    , where s is the amount of salt in gallons and t is time in minutes

for input I've tried

and for output

no luck yet, please help!

Bob

Administrator

Registered: 2010-06-20

Posts: 10,583

Re: Differential Equations - Brine Problem

hi b31den

Sorry not to have responded sooner.  I wasn't 100% about how I'd tackle this, so I waited to see someone else do it.

But no such luck, so I've had a think and this is my method.

Let S be the amount of impurities in the storage tank. (initially 5)

Then change in amount with time

So separate the variables and integrate from S = 5 to S = 0.5.  (are you OK to do this?)

I made it just over 85 mins.

To check my method, I have made a spreadsheet version with increments of 1 minute; calculating the amount of impurities removed and re-calculating how much is left for the next minute of processing.

This gave a close agreement with my integration method.

Hope that helps, 

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob