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

MATH102 December 2012

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Question C 04 (c)
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. (c) Sketch the graph of the solution to your differential equation showing that the mass of dead leaves tends toward a steady state assuming that initially (t = 0) there are no leaves on the ground.

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Draw the phase space to see the slopes of the solution. Then draw the solution by starting at $\ (0,0)$ and following the slopes until you reach the equilibrium.

Checking a solution serves two purposes: helping you if, after having used the hint, 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 sketch the solution without solving the equation. Hence, we draw the phase space for $\displaystyle {}Q$ (the number of leaves) and $\displaystyle {}{\textrm {d}}Q/{\textrm {d}}t$ (the daily rate of change of leaves). Therefore, we first make a table with $\ Q$ and the corresponding $\displaystyle {}dQ/dt$. $\ Q=0$ $\ Q=2$ $\ Q=4$ $\ Q=6$ $\ Q=8$ $\ Q=10$ $\ dQ/dt=5$ $\ dQ/dt=4$ $\ dQ/dt=3$ $\ dQ/dt=2$ $\ dQ/dt=1$ $\ dQ/dt=0$ After drawing the phase space, the curve of the solution can be plotted by following the slopes.

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