Problem 66 - Calculus for Biology and Medicine

Question

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t\) (measured in days). Assume that the radioactive decay rate of the material is \(0.2 /\) day. Find the differential equation for the radioactive decay function \(W(t)\).

Step-by-Step Solution

Answer

The differential equation is \(\frac{dW}{dt} = -0.2W\).

Step 1: Understanding Radioactive Decay

The process of radioactive decay can be described as a decrease in the amount of radioactive material over time, which typically follows an exponential pattern. The rate of decay is proportional to the quantity of the material present at any given time.

Step 2: Defining the Decay Rate Constant

The decay rate constant is given in the problem as \(0.2 /\text{day}\). This constant, typically denoted by \(k\), is used to express the proportion of the material that decays per unit time. In this case, \(k = 0.2\).

Step 3: Writing the Differential Equation

The general form of a first-order linear differential equation for radioactive decay is \(\frac{dW}{dt} = -kW\), where \(W\) is the amount of material and \(k\) is the decay rate constant. Here, \(k = 0.2\), so the differential equation becomes \(\frac{dW}{dt} = -0.2W\).

Step 4: Interpreting the Differential Equation

The equation \(\frac{dW}{dt} = -0.2W\) indicates that the rate of change of the radioactive material \(W(t)\) with respect to time \(t\) is proportional to the current amount of material and that it is decreasing over time with a rate constant of \(0.2\).

Key Concepts

Differential Equations

A differential equation is a mathematical equation that involves functions and their derivatives. In simple terms, it describes how a certain quantity changes over time. In the context of radioactive decay, differential equations help model the rate at which radioactive material decreases.

For radioactive decay, the differential equation often used is first-order and linear. This means it involves the first derivative of the function, which in this case, denotes how the amount of radioactive material, denoted as \( W(t) \), changes with time \( t \).

More succinctly, if \( \frac{dW}{dt} \) represents the rate of decay, this differential equation is expressed as:

\[ \frac{dW}{dt} = -kW \]

Here, \( k \) is a constant known as the decay rate constant. The negative sign indicates that the quantity of material is decreasing over time. Differential equations like these are useful because they allow us to predict the remaining quantity of radioactive substance at any given time.

Exponential Decay

Exponential decay refers to a process where a quantity reduces at a rate proportional to its current value. In simple terms, as more time passes, less of the substance remains, and its rate of reduction slows down. This kind of decay is common in radioactive materials.

For radioactive decay, the process follows an exponential function because the material decays at a rate proportional to its current amount. In mathematical terms, exponential decay can be represented as:

\[ W(t) = W_0 e^{-kt} \]

where:

• \( W(t) \) is the remaining amount of material at time \( t \)
• \( W_0 \) is the initial quantity of the material
• \( k \) is the decay rate constant
• \( e \) is the base of the natural logarithm, approximately equal to 2.718

As \( t \) increases, the product \( kt \) increases, leading to \( e^{-kt} \) becoming smaller, hence reducing \( W(t) \), following an exponential curve downwards.

Decay Rate Constant

A decay rate constant is a key aspect in the study of exponential decay processes, including radioactive decay. This constant, denoted as \( k \), quantifies the rate at which a quantity decays over time.

In our context, the decay rate constant is crucial as it dictates how quickly the radioactive substance reduces. The given value, \( k = 0.2 \) per day, means that each day, approximately 20% of the remaining material decays. This decay rate constant influences both the differential equation and the resulting exponential decay function.

The decay rate constant helps calculate how much of a radioactive material remains after a certain period, following the formula:

\[ \frac{dW}{dt} = -kW \]

This highlights its role in shaping the exponential decay pattern, allowing predictions on how long it takes for a material to reach a certain level. Understanding the decay rate constant also helps in practical applications, including medical treatments, archaeological dating, and nuclear energy management.