Hi, here is a very simple solution mixing problem that I can't solve which I am really ashamed of.(adsbygoogle = window.adsbygoogle || []).push({});

Problem. A vessel whose capacity is 5 liters contains 2 liters of 15% salt solution. How many liters of 20% salt solution have to be mixed to the 15% solution to produce a solution with as high salt concentration as possible?

My solution. I constructed this function for salt concentration in the new solution:

[tex]f(x)=\frac{0.3+0.2x}{2+x}[/tex]

(The amount of salt in the first solution is 2*0.15 = 0.3)

I obtained the derivative of the above function:

[tex]f'(x)=\frac{0.2(2+x)-(0.3+0.2x)}{\left(2+x\right)^2}=\frac{0.4+0.2x-0.3-0.2x}{\left(2+x\right)^2}=\frac{0.1}{\left(2+x\right)^2}[/tex]

I'm stuck at this point. If I try to find points of extremum, there are none. It means that I cannot increase salt concentration in the original solution by adding the said 20% salt solution. But I know the answer is 3 liters, so I must be doing something wrong.

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As a sidenote, this problem is taken from a nationalmathsexam. As such, you are not expected to think of mass, volume and density of water and salt. Also, I didn't make use of the fact that the vessel can contain up to 5 liters of liquid but just don't see where it fits in. Finally, I am not sure if this problem is even supposed to be solved using optimization techniques (derivatives and minima/maxima) but that was the first idea I got ...

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# A Simple Salt Solution Mixing Problem - yet stuck

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