Chapter 5.4, Problem 14E

Decay of Litter Litter such as leaves falls to the forest floor, where the action of insects and bacteria initiates the decay process. Let $A$ be the amount of litter present, in grams per square meter, as a function of time $t$ in years. If the litter falls at a constant rate of $L$ grams per square meter per year, and if it decays at a constant proportional rate of $k$ per year, then the limiting value of $A$ is $R=L/k$. For this exercise and the next, we suppose that at time $\text{t }=0$, the forest floor is clear of litter.

a. If $D$ is the difference between the limiting value and $A$, so that $D=R-A$, then $D$ is an exponential function of time. Find the initial value of $D$ in terms of $R$.

b. The yearly decay factor for D is $e^{-k}$. Find a formula for $D$ in term of $R$ and $k$. Reminder: ${\left(a^b\right)}^c=a^{bc}$.

c. Explain why $A=R-R{e}^{kt}$.