No more fear and loathing!

The problem.

Are you still here? Good. I like your spirit. Let's take a look at the problem.

The cooling system currently holds 4 quarts of antifreeze and 8 quarts of water.
I remove a small screw in the bottom of the radiator and let fluid drain out.
At the same time as the fluid drains out, I add pure antifreeze at the top.
Assume that I add pure antifreeze at the same rate as the mixture drains, so there is always exactly 12 quarts of fluid in the system.
Assume also that the antifreeze is mixed thoroughly and instantaneously as it is added.
How much fluid do I have to drain out and replenish with straight antifreeze in order to get a 50/50 mixture?

So, do you see the difficulty? The concentration of the fluid leaving the system keeps increasing over time.

The hint.

Want a hint? Here it is: If you were one of my students during the Fall of 2005, you damn well better know how to figure out the answer.

OK, that's a little over the top. Actually, I would consider the course a success if my students simply recognized that the problem involves differential equations. I suspect less than one in ten would be able to solve the problem, at least without a quick review. No knock on my students; it is just one of those courses one finds easy to forget.

The answer.

After all this, you want me to GIVE you the answer? Sure, no problem. It's a bit less than three and a half quarts.

What? You want to see my work? You don't trust me? Fine. I'll show you mine, if you show me yours.

Original Puzzler
How to Solve It
Challenge Problem
Differential Equations
Last modified:
© 2006
Peter C. Scott
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