Question

Calculus Help!! Dead leaves accumulate on the forest floor at a rate of 4 grams per square centimetre per year, while at the same time decomposing at a rate of 50% per year, Model this problem using a differential equation for Q(t), the amount of dead leaves in grams per square centimetre at time t, and solve the initial value problem with Q(0)=0

Answer 1

The best way to go about this is to assign differentials to the rate of accumulation on the ground and the rate of decomposition. The latter can be tricky but I suggest an exponential decay formula which may have the form of Ce^t.

Answer 2

Rate of leaves falling = 4
Rate of leaves decomposing = Q/2

\[\frac{dQ}{dt} = 4-\frac{Q}{2}\]
\[\frac{dQ}{dt} +\frac{Q}{2} = 4\]
Find integrating factor \[\rightarrow e^{\int\limits_{}^{}1/2} = e^{1/2t}\]
\[\frac{dQ}{dt}e^{1/2t} +\frac{Q}{2}e^{1/2t} = 4e^{1/2t}\]
\[(Qe^{1/2t})d/dt = 4e^{1/2t}\]
integrate both sides
\[Qe^{1/2t} = 8e^{1/2t} +C\]
\[Q = 8+Ce^{-1/2t}\]
solve for C using Q(0)=0
0=8+C --> C = -8
\[Q(t) = 8-8e^{-1/2t}\]

Answer 3

Thank you!!

Answer 4

welcome :)