Science:Math Exam Resources/Courses/MATH102/December 2012/Question C 04 (a)

MATH102 December 2012

  •  QA 1  •  QA 2  •  QA 3  •  QB 1  •  QB 2  •  QB 3  •  QB 4  •  QB 5  •  QB 6  •  QC 1  •  QC 2  •  QC 3(a)  •  QC 3(b)  •  QC 3(c)  •  QC 4(a)  •  QC 4(b)  •  QC 4(c)  •  QC 51  •  QC 52  •  

   Other MATH102 Exams

  •  December 2014  •  December 2011  •  December 2012  •  December 2013  •  December 2016  •  December 2015  •  

Question C 04 (a)
Dead leaves accumulate on the ground in a forest at a rate of 5 grams per square centimeter per year. At the same time, these leaves decompose at a continuous rate of 50 percent per year. (a) Write a differential equation for the total mass Q(t) of dead leaves (per square centimeter) at 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 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
You want to find an equation for the derivative of $\ Q$ with respect to time, ${\frac {d}{dt}}Q$, depending on $\ Q$.

Hint 2
The derivative ${\frac {d}{dt}}Q$ is a sum of two terms: The first term represents the leaves which are added to the ground every year, the second term presents the leaves which are removed from the ground every year. First, try to find these terms separately. Then add them and think about the sign they should have. Which of the terms depends on Q?

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 want to find the differential equation of the mass $\ Q$ of dead leaves. This means we find the derivative of $\ Q$ with respect to time, dependent on $\ Q$. Therefore, we separate $\ {\frac {d}{dt}}Q$ into $\ {\frac {d}{dt}}Q_{+}$ the leaves that are added every year and $\ {\frac {d}{dt}}Q_{-}$, that are removed every year. Such that ${\frac {d}{dt}}Q={\frac {d}{dt}}Q_{+}+{\frac {d}{dt}}Q_{-}$ The leaves (in grams) that are added every year per $\ cm^{2}$ are ${\frac {d}{dt}}Q_{+}=5{\frac {\text{grams}}{{\text{year}}\cdot cm^{2}}}.$ The leaves that are removed every year depends on the amount of leaves: According to the mass of leaves, half of the mass is lost every year: ${\frac {d}{dt}}Q_{-}=-{\frac {0.5Q}{\text{year}}}.$ Since leave mass is lost, we put a negative sign here. Hence, our final answer is ${\frac {d}{dt}}Q={\frac {5{\text{grams}}}{{\text{year}}\cdot cm^{2}}}-{\frac {0.5Q}{\text{year}}}.$ Note that this works out nicely so that the unit of Q is grams/cm2. With units suppressed, the above is ${\frac {d}{dt}}Q=5-0.5Q.$

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Population dynamics, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag