Problem 40 - Advanced Engineering Mathematics

Question

Radioactive Decay Suppose that $$ A^{\prime}(t)=-0.0004332 A(t) $$ represents a mathematical model for the decay of radium226 , 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

The radium sample weighs approximately 4.618 grams.

Step 1: Interpret the Given Information

The problem presents a differential equation \(A'(t) = -0.0004332 A(t)\), modeling the decay of radium-226. It also states the decay rate \(A'(t)\) is \(-0.002\) grams per year at a specific time \(t\). We need to find the amount of radium \(A(t)\) at this moment.

Step 2: Set Up the Equation for Specific Decay Rate

At the given rate, the equation becomes \(-0.002 = -0.0004332 A(t)\). This equation relates the decay rate to the amount of radium remaining \(A(t)\).

Step 3: Solve for \(A(t)\)

Rearranging the equation from Step 2, solve for \(A(t)\):\[A(t) = \frac{-0.002}{-0.0004332}.\]The negative signs cancel out, simplifying the calculation.

Step 4: Calculate the Amount of Radium

Perform the division:\[A(t) \approx \frac{0.002}{0.0004332} \approx 4.6178.\]This means there are approximately 4.618 grams of radium remaining at the time when the decay rate is \(-0.002\) grams per year.

Key Concepts

Differential Equations

Differential equations are a powerful tool used to model situations where quantities change. They are equations that involve functions and their derivatives. In the context of radioactive decay, differential equations allow us to express how the quantity of a radioactive substance decreases over time.
A differential equation, like the one presented in the exercise, shows the relationship between the rate of change of the amount of a substance at any given time and the amount itself. Specifically, in this case, the differential equation is:

• \( A'(t) = -0.0004332 A(t) \)

This equation tells us how the derivative of the amount of radium (\( A(t) \)), which represents its rate of change over time, is proportional to the amount itself. The negative sign indicates that the amount is decreasing.
Understanding and solving such equations permits us to predict how much of a radioactive substance will remain after a given time. In practice, solving a differential equation involves integration, which is the inverse process of differentiation.

Radium-226 Decay

Radium-226 is a naturally occurring radioactive isotope that undergoes radioactive decay. It is known for its long half-life of about 1600 years, meaning it takes this amount of time for half of a given amount of radium-226 to decay.
The decay process follows an exponential decay model. This model reflects how radium-226 diminishes over time. The equation provided in the exercise is a mathematical representation of this type of decay process.
Radioactive decay processes, like that of radium-226, are crucial in various fields, including medicine for cancer treatment, archaeology for dating artifacts, and geology for understanding geological formations. Understanding the decay of radium-226 can also help with handling materials that pose radiation hazards.
The presented decay model:

• \( A'(t) = -0.0004332 A(t) \)

illustrates how radium-226 decays proportionally to its current quantity, not in a linear fashion, but rather exponentially. This intrinsic property leads to a more rapid initial decay rate when more substance is present, slowing down as the quantity decreases.

Decay Rate Calculations

Calculating the decay rate involves analyzing how quickly the substance's amount is decreasing at a specific moment. This requires understanding the relationship defined by the differential equation.
In this exercise, you are dealing with finding the amount of radium when the decay rate is a known value, namely \( -0.002 \) grams per year. By substituting \( A'(t) = -0.002 \) into the equation from the differential equation, we can solve for \( A(t) \). This involves a simple algebraic rearrangement:

• Set \( -0.002 = -0.0004332 A(t) \)
• Rearrange to solve for \( A(t) \): \( A(t) = \frac{-0.002}{-0.0004332} \)

By performing this division, you determine the amount of radium present at the time when the decay rate is at the specified value. This process allows us to ascertain how much radium remains without tracking it continuously over time.
Practical decay rate calculations like this one have significant applications, helping to predict long-term behavior of substances, manage radioactive materials safely, and aid scientists in estimating long-term effects of radioactive decay in environmental studies.