Science:Math Exam Resources/Courses/MATH101/April 2013/Question 08
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Question 08 

FullSolution Problems. In questions 4–12, justify your answers and show all your work. If a box is provided, write your final answer there. Unless otherwise indicated, simplification of numerical answers is required in these questions. A tank contains 1000 L (litres) of pure water. A solution that contains 0.01 kg/L of sugar is poured into the tank at a rate of 20 L/min. The tank's contents are throughly mixed and drain out of the tank at the same rate. How much sugar is in the tank after one hour? 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Set up what you know and what you want in variables. What is the relevant equation which relates the variables and quantities? 
Hint 2 

Don't forget the absolute value in . How does this affect isolating for your desired variable? 
Hint 3 

What are the initial conditions, and how does that affect your solution? 
Hint 4 

Be careful about your units! 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Let be the mass of sugar in the tank at time (in minutes), with units of kg. The rate of change of is the difference between the incoming rate of salt and the outgoing rate of salt. The rate is simply the concentration multiplied by the flow rate; for incoming, we know the concentration is 0.01 kg/L and the flow rate is 20 L/min; for the outgoing, the concentration is and the flow rate is still 20 L/min. If , then We have an initial condition of , so (Incidentally, this initial condition implies that , since is a steadystate solution, where the mass never changes from 10.) Now we solve Since is continuous, , and the righthand side of the equation above is always positive, we conclude for all . Then The amount of sugar in the tank after 1 hour, or 60 minutes, is 