Science:Math Exam Resources/Courses/MATH101/April 2011/Question 05 (a)

MATH101 April 2011

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Question 05 (a)

Bird-Bath and Beyond Incorporated is famous for its Quetzal attracting bird-feeder solution made from water, honey and cane-sugar. To make their solution, both honey and a cane-sugar solution are poured into a 200 L mixing tank. The honey is poured in at a rate of 1 L per minute while the sugar solution is poured in at 9 L per minute. Note that 1 L of honey contains 1 kg of sugar, while the cane-sugar solution contains 100g of sugar per L. Unfortunately today there is a problem with the mixing tank. It was thoroughly cleaned and is initially filled with pure water, but main valve was broken and the water cannot be drained. When the mixing process is started, the honey and sugar-solutions are poured into the tank, and the excess fluid flows out of an emergency valve at 10 L per minute and onto the floor. You should assume that the solutions mix immediately and thoroughly in the tank. (a) Give a differential equation that is satisfied by the amount of sugar in the tank after time t.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
How much sugar (in kg) is going inside of the tank every minute? How much sugar (in kg) is going out of the tank every minute?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We will denote by S(t) the amount of sugar in the tank at time t (where t is in minutes). The differential equation that we are building should look like S'(t) = (rate at which sugar is added to the tank) - (rate at which sugar is spilled out of the tank) Every minute, there is 1 L of honey poured in the tank, that litre contains 1kg of sugar; and 9 L of sugar solution are also poured in, containing 900 g of sugar (since each litre of sugar solution contains 100 g of sugar). Hence all in all, 1.9 kg of sugar is poured in the tank every minute. Now, we still have to deal with the water that is spilling on the floor. The question asks us to assume that the sugar solution and honey are instantly mixed to the rest of the tank, so when dealing with the fluid going out of the tank, we can assume it has the same concentration of sugar as in the rest of the tank. What is that concentration? Well there are S(t) kilograms of sugar in the tank at that time and since the tank is full, there is 200 L of liquid in it, so the concentration of sugar in the tank at time t is S(t)/200 kg/L. Since we are told that the fluid is spilling at an instantaneous rate of 10 L per minute, this means that every minute, there are $10\cdot {\frac {S(t)}{200}}={\frac {1}{20}}S(t)$ kilograms of sugar spilled out of the tank. Putting these two bit of information together, we obtain that the amount of sugar in the tank must satisfy the differential equation ${\frac {dS}{dt}}=1.9-{\frac {1}{20}}S(t).$

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