Science:Math Exam Resources/Courses/MATH110/December 2016/Question 09 (c)
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Question 09 (c) 

Sugar dissolves in water at a rate proportional to the amount still undissolved. Suppose the amount of undissolved sugar present in a water solution hours after the sugar was added to the water is modelled using an exponential decay function , where and are positive constants. Suppose initially of sugar are added to the water. Assume it takes for half of the initial amount of sugar to dissolve. Answer the questions below. You may leave your answers unsimplified, i.e. in ”calculatorready” form. (c) If 10 more kg of sugar is added to the solution after the solution was first mixed, when will there be of undissolved sugar left in the solution? 
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 

We need to determine the unknown constants again. 
Hint 2 

The constant remains unchanged because it indicates how fast sugar dissolves. 
Hint 3 

The constant is changed. It is now determined by the amount of sugar at time . 
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. Following the Hints, let us denote the amount of undissolved sugar after by According to the description, we have that Putting , So the new formula for the amount of undissolved sugar is Now we solve for if : Taking the natural logarithm on both side, Answer: there will be kg of sugar at time hours. 