Math Ext 1 - Got Some Questions (1 Viewer)

Life'sHard

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Hey guys, was feeling bored and decided to create a math ext 1 thread for everyone to contribute in posting questions here. Hopefully, some brilliant minds out there will be able to solve your questions.

This thread is for anyone who needs help. Or anyone bored of studies to do some questions :)
 

jimmysmith560

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This question is from the James Ruse Agricultural High School Year 12 Mathematics Extension 1 2020 Trial HSC Examination:

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CM_Tutor

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We know that water is entering at 3 L per min and being removed at 2 L per min. It follows that


And hence, the volume, as a function of time, is given by:


Taking the origin of time, , as the "initial" point when there was of salt in of water:


Now, the quantity of salt at time is changing as salt is removed from the tank and water is added, and thus:

 

Trebla

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By conservation of mass we must have that



However, since there is no salt coming into the tank (i.e. ) the problem now reduces to figuring out how much salt is coming out the tank at time t, so


where V is the volume of water.

The concentration C in the tank is the same as the concentration of the water coming out at an instantaneous time t (note this is slightly "hand-wavey" in terms of rigour, the Ruse solution has a bit more rigour by using differentials though that is a tad outside of the syllabus), so



But by definition



We know that the tank's net change of volume per minute is given by


since the volume is initially at 10L so c = 10.

Substitute it all in to get

 

CM_Tutor

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These tank problems can be expanded into quite challenging longer questions. Here's an example to ponder (no exam would ask all of this at once, but to show what's possible):

A tank that can hold 100 L of liquid is designed with a built-in stirrer to thoroughly mix its contents and with a tap to allow fluid to be removed at rates up to six litres per minute. The tank is initially filled with a solution in which kg of sugar is dissolved. At time minutes, the tap is fully opened and simultaneously pure water is added to the tank at a rate of four litres per minute.

Let kg be the mass of sugar in the tank at time min. Also at time min, let kg/L be the concentration of sugar in the tank, and let L be the volume of solution in the tank. It follows that , , and are related by the equation .



















 

CM_Tutor

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I still believe there should be an easier way to do this. So hoping someone else can decipher this mess of working. @CM_Tutor hahahaha
For an MCQ, you can shortcut the process. If there is only water coming in then the time derivative will be proportional to the rate that solution is leaving, proportional to the amount at time t, and inversely proportional to the volume. This can be seen from the units:

Solution leaving is in L per min (say), quantity is in kg, volume is in L, so you get


Rate of outflow is negative and the volume may be increasing or decreasing.

You should be able to eliminate the incorrect options in an MCQ fairly easily, and to check if you have to do the calculation whether your result is reasonable in form if the question is not in "Show that ..." form.

If the solution entering has solute in it, you'd need to add a term for rate of inflow (in L / min) time concentration in inflow (in kg / L) to get a change term in kg / min.

@B1andB2, you now have the working from the approach used in the solutions of the trial, the approach I use, and the one Trebla has provided... does this make sense now?
 

B1andB2

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For an MCQ, you can shortcut the process. If there is only water coming in then the time derivative will be proportional to the rate that solution is leaving, proportional to the amount at time t, and inversely proportional to the volume. This can be seen from the units:

Solution leaving is in L per min (say), quantity is in kg, volume is in L, so you get


Rate of outflow is negative and the volume may be increasing or decreasing.

You should be able to eliminate the incorrect options in an MCQ fairly easily, and to check if you have to do the calculation whether your result is reasonable in form if the question is not in "Show that ..." form.

If the solution entering has solute in it, you'd need to add a term for rate of inflow (in L / min) time concentration in inflow (in kg / L) to get a change term in kg / min.

@B1andB2, you now have the working from the approach used in the solutions of the trial, the approach I use, and the one Trebla has provided... does this make sense now?
yeah, thanks a lot! Are these type of concentration questions something we could expect to be in the HSC?
 

CM_Tutor

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yeah, thanks a lot! Are these type of concentration questions something we could expect to be in the HSC?
Tank problems with constant volume have been asked for ages. The ones with variable volume have been appearing in trials under the new syllabus, so I think you need to have tried them. The expanded question that I posted is then using them in differential equations questions that are certainly within the new syllabus.

Will the HSC ask one? I think they could... if I knew for sure what was on this year's HSC, I couldn't tell anyone! :)

... and for the record, I don't have any inside information on the content of the HSC exams beyond what is publicly known.
 

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