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

MATH110 December 2016

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Question 09 (a)

Sugar dissolves in water at a rate proportional to the amount still undissolved. Suppose the amount $M$ of undissolved sugar present in a water solution $t$ hours after the sugar was added to the water is modelled using an exponential decay function $M(t)=Ce^{-kt}$, where $C$ and $k$ are positive constants. Suppose initially $10{\text{ kg}}$ of sugar are added to the water. Assume it takes $4{\text{ hr}}$ for half of the initial amount of sugar to dissolve. Answer the questions below. You may leave your answers unsimplified, i.e. in ”calculator-ready” form. (a) Find the exponential function that predicts the amount of undissolved sugar $M(t)$ present in the solution at $t{\text{ hr}}$. Make sure your answer has no unknown constants.

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
Use the given conditions to find the unknown constants first.

Hint 2
The word "initially" means ${\textstyle t=0}$.

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

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 plug in the given conditions into the formula ${\textstyle M(t)=Ce^{-kt}}$ (unit: kg). Since initially the amount is ${\textstyle 10}$ kg, we have $$M(0)=10.$$ This means $$Ce^{-k\cdot 0}=10,$$ or $$C=10.$$ Now that ${\textstyle M(t)=10e^{-kt}}$, we plug in the second condition that when ${\textstyle t=4}$, ${\textstyle M(t)={\dfrac {1}{2}}M(0)}$. So $$10e^{-k\cdot 4}={\dfrac {1}{2}}\cdot 10,$$ $$e^{-4k}={\dfrac {1}{2}}.$$ Raising both sides to the power ${\textstyle {\dfrac {1}{4}}}$, we have $$e^{-k}=({\dfrac {1}{2}})^{\frac {1}{4}}=2^{-{\frac {1}{4}}}.$$ This gives $$M(t)=10(2^{-{\frac {1}{4}}})^{t}=10\cdot 2^{-{\frac {t}{4}}}.$$ Answer: ${\textstyle {\color {blue}M(t)=10\cdot 2^{-{\frac {t}{4}}}}.}$