91Ó°ÊÓ

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 \(t=0\), 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 terms of \(R\) and \(k\). Reminder: \(\left(a^{b}\right)^{c}=a^{b c}\). c. Explain why \(A=R-R e^{-k t}\).

Step by step solution

01

Setting up initial conditions and expressions

We start with the information given and determine initial expressions. Since at time \(t=0\), the forest floor is clear of litter, it means that \(A(0) = 0\). The limiting value of \(A\) is \(R = \frac{L}{k}\), and \(D = R - A\). Initially, the difference \(D(0) = R - A(0) = R\). Thus, the initial value of \(D\) is simply \(R\).

02

Expressing D as an exponential function

To find the formula for \(D\), we use the information that the yearly decay factor for \(D\) is \(e^{-k}\). This means \(D\) decays exponentially. We relate the decay: \(D(t) = D(0) \, e^{-kt}\). Given \(D(0) = R\), we get \(D(t) = R \, e^{-kt}\).

03

Connecting A to D for the given form

We know \(D = R - A\). Substituting the expression for \(D(t)\) from Step 2 into \(D=R-A\), we get: \(R - A(t) = R \, e^{-kt}\). Solving for \(A(t)\), we find: \(A(t) = R - R \, e^{-kt}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decay Process

The decay process is common in nature and describes how substances break down over time. In this context, it refers to the decomposition of forest litter like leaves, facilitated by insects and bacteria. This process can be modeled mathematically using exponential functions.
Typically, decay involves a constant rate at which the substance reduces. For the leaves on the forest floor, they decay at a constant proportional rate, denoted by a decay constant \(k\).
Understanding this concept helps in predicting how much litter will remain after a certain period, given the starting conditions and the rate of decay.

Limiting Value

The limiting value is a concept in exponential decay processes that represents the long-term stable amount of a substance. It is the value that the function approaches as time progresses towards infinity. In the litter example, this is the steady state amount of litter present when the rate of accumulation equals the rate of decay.
Mathematically, the limiting value \(R\) can be calculated using the ratio \(\frac{L}{k}\), where \(L\) represents the constant rate at which litter falls, and \(k\) is the decay constant. As \(t\) becomes large, the difference between the current amount \(A\) and \(R\) diminishes, signifying that \(A\) is reaching its limiting value.
The limiting value helps in understanding the equilibrium state in a decay process and is crucial for modeling long-term behavior of dynamic systems.

Exponential Decay

Exponential decay describes a process whereby the quantity decreases at a rate proportional to its current value. This concept is represented using exponential functions and often features in fields such as physics, biology, and environmental science.
In our exercise, the quantity \(D\), which is the difference between the limiting value \(R\) and the current amount \(A\), decays exponentially over time. The formula \(D(t) = R \, e^{-kt}\) highlights this behavior, showing that \(D\) decreases at a rate tied to the decay factor \(e^{-k}\).
Recognizing exponential decay is crucial when modeling situations where a constant percentage of the amount decreases over time, enabling accurate predictions of substance behavior.

Initial Conditions

Initial conditions are critical in solving differential equations and understanding the behavior of dynamic systems. They represent the state of a system at the starting point of analysis, often used as a reference for future predictions.
In this exercise, the initial condition is that at \(t=0\), the forest floor is clear of litter, meaning \(A(0) = 0\). This initial condition helps establish the formula for the difference \(D\) and solve for \(A\) over time.
Understanding initial conditions allows us to trace the evolution of a system accurately and tailor the general equations to specific scenarios, ensuring that our mathematical models reflect real-world situations.