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 \( \frac{dW}{dt} = -0.2W(t) \).

Step 1: Understanding The Decay Rate

The problem states that the decay rate of the material is constant at 0.2 per day. This means that the amount of substance decreases by 20% each day.

Step 2: Identify The General Form of Decay

Since the decay process is continuous and exponential, the differential equation generally takes the form \( \frac{dW}{dt} = -kW(t) \), where \( k \) is the decay constant.

Step 3: Substitute The Decay Constant

We know the decay rate is 0.2 per day, so we set \( k = 0.2 \). Substitute \( k \) into the differential equation to get \( \frac{dW}{dt} = -0.2W(t) \).

Step 4: Verify The Differential Equation

To ensure the correctness of the differential equation, observe that the negative sign indicates decay, meaning the substance amount decreases over time, aligning with the physical context of radioactive decay.

Key Concepts

Radioactive Decay

Radioactive decay is a process where unstable atomic nuclei lose energy by emitting radiation. This phenomenon is described using differential equations to model how the quantity of radioactive material changes over time. The material, often called a radioactive isotope, decreases at a rate that is proportional to its current amount.

In the context of radioactive decay, the amount of material left after time, denoted as \(W(t)\), follows a predictable pattern. The change in the amount over time \(\left( \frac{dW}{dt} \right)\) is negative because the material is continuously decreasing. This decay process is exponential due to the constant proportion of material lost over equal time periods.

Key features of radioactive decay involve:

• Understanding that decay is a random process at the atomic level but predictable in large quantities.
• The decay rate, constant for each material, determines how fast the decay happens.
• Applications such as carbon dating or medical treatments rely on understanding this decay behavior.

Exponential Decay

Exponential decay is a process where the amount of some entity decreases at a rate proportional to its current value. In mathematical terms, this relationship is often expressed as \(\frac{dY}{dt} = -kY(t)\), where \(Y(t)\) is the quantity that decays exponentially, and \(-k\) is the decay constant.

The nature of exponential decay means that the larger the quantity present, the greater the amount that is lost over a given interval. This is seen in the case of radioactive materials where each day, 20% of the current amount is lost, leading to a rapid decrease initially, followed by a more gradual decline.

Characteristics of exponential decay include:

• The decay is rapid at first when quantity is large and slows down as quantity decreases.
• The half-life of a substance, the time it takes for half of the material to decay, is a direct consequence of this decay pattern.
• It models not just physical processes but also financial, biological, and other natural occurrences.

Decay Rate

The decay rate in the context of differential equations and radioactive decay is crucial. It defines how quickly the substance loses its quantity over time. For radioactive materials, this rate is often given as a constant proportion per time unit, like the 0.2 per day for a particular isotope in our problem.

Understanding decay rate helps in:

• Predicting how long a substance takes to decrease to a desired amount.
• Conversely, calculating the initial amount if the end amount and time are known.
• Estimating lifespan and effectiveness of materials in various applications.

Integrating this decay rate into the differential equation \(\frac{dW}{dt} = -0.2W(t)\) helps us predict the behavior of the substance over time. The negative sign in the equation confirms that the substance is losing mass, consistent with the physical phenomena observed in radioactive decay.