Problem 67 - 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 \(4 /\) day. Find the differential equation for the radioactive decay function \(W(t)\).

Step-by-Step Solution

Answer

The differential equation is \( \frac{dW}{dt} = -4W \).

Step 1: Understanding Radioactive Decay

Radioactive decay follows an exponential decay model, where the rate of decay is proportional to the amount of material present.

Step 2: Expressing the Decay Proportionally

Since the decay rate is proportional to the remaining amount, we express this as \( \frac{dW}{dt} = -kW \), where \( k \) is the decay constant.

Step 3: Identifying the Decay Constant

We are given that the decay rate is "4 per day." This implies that \( k = 4 \).

Step 4: Writing the Differential Equation

Substitute \( k = 4 \) into the proportional decay equation to get \( \frac{dW}{dt} = -4W \). This is the differential equation for the decay function \( W(t) \).

Key Concepts

Radioactive Decay

Radioactive decay is a natural process by which unstable atomic nuclei lose energy by emitting radiation. This process results in the transformation of an element into a different element or a different isotope of the same element. A key characteristic of radioactive decay is its random nature, and at its heart lies the principle of probability.

In radioactive decay, the way we measure how quickly a sample decays is through its decay rate. This decay rate is often described in terms of a decay constant, which gives the proportional relationship between the number of radioactive atoms and how quickly they decay.

• The decay rate indicates how much of the radioactive material remains over a period of time.
• Each radionuclide has a specific decay constant that identifies how fast it decays.
• The decay process is not affected by physical conditions like temperature or pressure since it's an intrinsic property of the radioactive atom.

This decay rate, as described mathematically, can be represented through a differential equation which helps us determine how the quantity of radioactive substance decreases over time.

Exponential Decay

Exponential decay is a process where the quantity of an item decreases at a rate that is proportional to its current value. This means the rate of decrease is quick when the quantity is large and slows down as the quantity decreases. Exponential decay can be seen in processes like radioactive decay as well as in other contexts such as depreciation of assets or cooling of a hot object.

The mathematical representation of exponential decay is often expressed through the formula:\[ N(t) = N_0 \cdot e^{-kt} \]Here:

• \( N(t) \) represents the amount remaining at time \( t \).
• \( N_0 \) is the initial amount.
• \( k \) is the decay constant and it dictates the rate of decay.
• \( e \) is the base of the natural logarithm, approximately equal to 2.718.

This formula underscores that the decay process is exponential, meaning it decreases in a multiplicative manner. In the context of radioactive decay and other similar processes, understanding exponential decay is key to predicting and analyzing the rate of decline of a given quantity over time.

Calculus

Calculus is a fundamental branch of mathematics that deals with rates of change and the accumulation of quantities. It is a tool that allows us to systematically analyze changes and predict future values. In the context of exponential decay, calculus helps us model and solve differential equations which are crucial in understanding the underlying behavior of changing quantities over time.

One of the essential tools in calculus is the differential equation. Differential equations involve derivatives, which represent the rate of change of a variable with respect to another. In the case of radioactive decay, the differential equation can be expressed as:\[ \frac{dW}{dt} = -kW \]Here:

• \( \frac{dW}{dt} \) signifies the instantaneous rate of change of the amount of substance \( W \) with respect to time \( t \).
• \( k \) is the decay constant and indicates the proportionality between the rate of decay and the current amount of substance.

This equation showcases how calculus is employed to predict how a radioactive material or any decaying quantity behaves over time. Through integration and differential equations, calculus allows us to describe the entire process and make quantitative predictions about states in the future.