Science:Math Exam Resources/Courses/MATH102/December 2012/Question C 04 (a)
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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 with respect to time, , depending on . 
Hint 2 

The derivative 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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. We want to find the differential equation of the mass of dead leaves. This means we find the derivative of with respect to time, dependent on . Therefore, we separate into the leaves that are added every year and , that are removed every year. Such that
The leaves (in grams) that are added every year per are
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:
Since leave mass is lost, we put a negative sign here. Hence, our final answer is
Note that this works out nicely so that the unit of Q is grams/cm^{2}. With units suppressed, the above is
