Problem 73 - Calculus for Biology and Medicine

Question

Radioactive Decay Suppose \(W(t)\) denotes the amount of a radioactive material left after time \(t .\) Assume that \(W(0)=5\) and \(W(1)=2\) (a) Find the differential equation that describes this situation. (b) How much material is left at time \(t=3\) ? (c) What is the half-life of the material?

Step-by-Step Solution

Answer

(a) \( \frac{dW}{dt} = -0.916W \). (b) \( W(3) \approx 0.488 \). (c) Half-life \( \approx 0.756 \text{ hours} \).

Step 1: Understand the Exponential Decay Model

Radioactive decay is typically modeled with an exponential decay function. This means we assume the function for the remaining material, \( W(t) \), can be expressed as \( W(t) = W(0) e^{-kt} \), where \( k \) is a positive constant representing the decay rate, and \( t \) is time.

Step 2: Determine the Decay Constant

We know at \( t=0 \), \( W(0) = 5 \). At \( t=1 \), \( W(1) = 2 \). Use these to find \( k \):\[ W(1) = W(0) e^{-k(1)} = 2 \]\[ 5e^{-k} = 2 \]Solve for \( k \):\[ e^{-k} = \frac{2}{5} \]\[ -k = \ln \left(\frac{2}{5}\right) \]\[ k = -\ln \left(\frac{2}{5}\right) \approx 0.916 \]

Step 3: Write the Differential Equation

Given \( W(t) = W(0)e^{-kt} \) and \( W(0) = 5 \), the differential equation that describes the rate of change of \( W \) is:\[ \frac{dW}{dt} = -kW \]Plugging \( k \) back:\[ \frac{dW}{dt} = -0.916W \]

Step 4: Calculate the Amount Left at \( t=3 \)

With \( k = 0.916 \), the equation \( W(t) = 5e^{-0.916t} \) helps find \( W(3) \):\[ W(3) = 5e^{-0.916 \times 3} \approx 0.488 \]

Step 5: Find the Half-Life of the Material

The half-life \( t_{1/2} \) is the time where \( W(t) = \frac{W(0)}{2} \). Set \( W(t_{1/2}) = \frac{5}{2} \):\[ \frac{5}{2} = 5e^{-0.916t_{1/2}} \]\[ e^{-0.916t_{1/2}} = \frac{1}{2} \]\[ -0.916t_{1/2} = \ln \left(\frac{1}{2}\right) \]\[ t_{1/2} = \frac{\ln \left(\frac{1}{2}\right)}{-0.916} \approx 0.756 \text{ hours} \]

Key Concepts

Exponential Decay Model

The Exponential Decay Model is a mathematical representation frequently utilized to describe the process of radioactive decay. In this model, the quantity of a substance decreases exponentially over time. The formula for exponential decay is typically expressed as:
\[ W(t) = W(0) e^{-kt} \]

• \(W(t)\) represents the amount of material remaining at time \(t\).
• \(W(0)\) is the initial amount of the material when \(t = 0\).
• The constant \(k\) is known as the decay constant and indicates the rate at which the material decays.

The key insight here is that in such decay scenarios, the material reduces at a rate proportional to its current amount, making it ideal for describing phenomena like radioactive decay.

Decay Constant

The Decay Constant \(k\) is a crucial factor in determining how quickly a radioactive substance diminishes over time. In the context of our exercise, it tells us the speed of decay:
- A larger \(k\) implies a faster decay.- A smaller \(k\) indicates a slower decay.
To calculate \(k\), use the known values of the material at two different times. From the exercise:
- At \(t = 0\): \(W(0) = 5\)- At \(t = 1\): \(W(1) = 2\)
Plug the values into the decay formula to solve for \(k\):
\[5e^{-k} = 2\]
Solving yields \( k \approx 0.916 \). This constant quantifies the decay per unit time, offering insights into how rapidly the substance loses its radioactivity.

Half-life

Half-life is the period it takes for half of the radioactive material to decay. It's a useful measure to easily estimate how long a material will remain active:
- The half-life is independent of the initial amount.- Defined by the relation where the remaining material \(W(t)\) equals half the initial amount \(W(0)\).
In our exercise, since \(W(t_{1/2}) = \frac{5}{2}\), it can be calculated through:
\[ e^{-0.916t_{1/2}} = \frac{1}{2} \]Consequently, solving for \(t_{1/2}\):\[ t_{1/2} = \frac{\ln \left(\frac{1}{2}\right)}{-0.916} \approx 0.756 \text{ hours} \]This value means that approximately every 0.756 hours, the amount of radioactive material will be halved, which provides important information for handling and safety.

Differential Equation

In radioactive decay, the relationship of how material diminishes over time can be described using a differential equation. This concept is fundamental for modeling decay processes:
The differential equation representing this is:
\[ \frac{dW}{dt} = -kW \]
This equation states:

• The rate of change of \(W\) over time \(t\) is proportional to the current amount \(W\).
• \(-k\) signifies the decay constant, which is negative to indicate the decrease.

Plugging in the decay constant from our calculations, we get:
\[ \frac{dW}{dt} = -0.916W \]
This equation offers a continuous depiction of how quickly the material's quantity is changing moment to moment, thus providing a powerful tool to predict future states of the material.