Problem 40 - Advanced Engineering Mathematics

Question

Suppose that $$ A^{\prime}(t)=-0.0004332 A(t) $$ represents a mathematical model for the decay of radium 226 , where \(A(t)\) is the amount of radium (measured in grams) remaining at time \(t\) (measured in years). How much of the radium sample remains at time \(t\) when the sample is decaying at a rate of \(0.002\) grams per year?

Step-by-Step Solution

Answer

Approximately 4.617 grams of radium remains.

Step 1: Identify the given rate of decay

The problem specifies that the decay rate at time \( t \) is \( 0.002 \) grams per year. This means \( A'(t) = -0.002 \).

Step 2: Use the given differential equation

The mathematical model given is \( A'(t) = -0.0004332 \cdot A(t) \). We need to find \( A(t) \) using this differential equation model.

Step 3: Set the decay rate equal to the model equation

We equate the given decay rate to the model equation: \(-0.002 = -0.0004332 \cdot A(t)\).

Step 4: Solve for \( A(t) \)

To find \( A(t) \), rearrange the equation from the previous step: \[ A(t) = \frac{0.002}{0.0004332} \].

Step 5: Calculate the value of \( A(t) \)

Perform the division \( \frac{0.002}{0.0004332} \) to find \( A(t) \). Calculating this gives \( A(t) \approx 4.617 \) grams.

Key Concepts

Radioactive Decay

Radioactive decay refers to the process by which an unstable radioactive element loses its energy over time by emitting radiation. For instance, radium 226 decays naturally and its amounts reduce at a predictable rate. This phenomenon is vital in fields such as physics, chemistry, and environmental science.

Let's break down the concept:

• "Radioactive" describes atoms that decay naturally and spontaneously.
• "Decay" implies a reduction in the number of radioactive atoms over time.
• With radium 226, this natural decay process releases energy in the form of radiation.

Understanding radioactive decay allows us to:

• Predict how many atoms will remain after a given amount of time.
• Determine the amount of radiation emitted at any moment.

In our exercise, the equation \( A'(t) = -0.0004332 \cdot A(t) \) models how the quantity of radium decreases over time.

Exponential Decay

Exponential decay is a mathematical concept where the quantity decreases at a rate proportionate to its current value. This characteristic is shown in various natural phenomena, including radioactive processes.

The key elements of exponential decay are:

• The decay rate, which signifies how fast the quantity reduces.
• The exponential function itself, which describes the general form of how quantities shrink rapidly then slow down.

In the context of our problem, the decay rate is expressed by the coefficient \(-0.0004332\) in the differential equation. This implies that the decrease in radium follows a pattern where each quantity reduction is determined by the existing amount.
The mathematical representation of exponential decay helps us:

• Forecast the long-term behavior of the substance.
• Determine specific milestones, such as how much of a sample remains at a particular point in time.

Solving these types of equations allows scientists to estimate how long a substance will be effective or potentially dangerous.

Modeling Real-World Problems

Modeling real-world problems are essential in understanding how various phenomena occur and change over time. By using differential equations, we can capture these processes mathematically.

A mathematical model is a representation of a physical process using mathematical concepts and language. It allows us to simulate and analyze behaviors and predict future states. For example:

• In the case of radium 226, the differential equation \( A'(t) = -0.0004332 \cdot A(t) \) models the decay process.
• It ties the decay rate to the current amount, giving insight into how quickly the sample decreases.

The benefits of mathematical modeling include:

• Allowing decision-makers to test scenarios and make informed predictions.
• Helping researchers design safe handling and disposal practices for radioactive materials.

Our exercise offers a practical example of how mathematics helps translate complex processes into more manageable and understandable formats. This empowers scientists, engineers, and other professionals to build more accurate and reliable models for various applications.