Problem 6 - Advanced Engineering Mathematics

Question

Initially, 100 milligrams of a radioactive substance was present. After 6 hours the mass had decreased by \(3 \%\). If the rate of decay is proportional to the amount of the substance present at time \(t\), find the amount remaining after 24 hours.

Step-by-Step Solution

Answer

About 88.24 mg remains after 24 hours.

Step 1: Understand Exponential Decay

The decay of a substance is modeled by exponential decay, which can be described by the equation \(A(t) = A_0 e^{-kt}\), where \(A_0\) is the initial amount, \(k\) is the decay constant, and \(t\) is time in hours. We start with an initial amount \(A_0 = 100\) mg.

Step 2: Determine the Decay Constant

Given that the mass decreases by 3% in 6 hours, the mass at 6 hours is \(97\%\) of the initial amount. Thus, \(A(6) = 97\). Set the equation \(97 = 100e^{-6k}\) to solve for \(k\).

Step 3: Solve for Decay Constant k

Rearrange the equation to \(e^{-6k} = \frac{97}{100}\). Taking the natural logarithm of both sides gives \(-6k = \ln(\frac{97}{100})\). Solve for \(k\) by dividing by \(-6\): \(k = -\frac{\ln(\frac{97}{100})}{6}\).

Step 4: Find the Amount Remaining After 24 Hours

Plug \(k\) back into the decay equation and solve for \(A(24)\): \(A(24) = 100 e^{-24k}\). Substitute the value of \(k\) and calculate the remaining amount.

Step 5: Calculate the Final Amount

With \(k\) calculated, substitute into \(A(24) = 100e^{-24(-\frac{\ln(\frac{97}{100})}{6})}\) to find \(A(24)\). This gives \(A(24) \approx 88.24\) mg.

Key Concepts

Radioactive Decay

Radioactive decay is an intriguing natural process where an unstable atomic nucleus loses energy by emitting radiation. Simply put, it's like the nucleus is shedding extra weight. The rate at which this shedding happens is not random; rather, it's consistent and follows a predictable pattern known as exponential decay. When something decreases exponentially, it shrinks proportionally to its size. In our example, a substance begins with 100 mg. As time passes, it gradually transforms or decays into something new, losing parts of itself as energy. By knowing the starting amount and the rate of decay, we can use math to predict how much will be left at any future point. In this case, math helps us solve how much of the original substance remains after any given time, like 24 hours in our problem.

Half-Life

Half-life is a term often used to describe the time required for a quantity to reduce to half its initial value. This concept is crucial in understanding radioactive decay because it gives us a snapshot of the decay rate over time. For example:

• If a substance's half-life is 6 hours, then in 6 hours the quantity will be reduced to half its original mass.
• By knowing the half-life, we can estimate how much of a substance remains at any given time by simple calculations, even if the decay rate isn't a perfect 50% every half-life.

Although the half-life isn't directly calculated in the exercise, understanding it helps rationalize why only 3% of the substance decayed in 6 hours instead of half. It enriches our understanding of how quickly or slowly a substance actually decays.

Decay Constant

The decay constant, often represented by the symbol \(k\), is a critical part of the exponential decay equation. It serves as the unique identifier of how fast a radioactive material decays. In our exercise, we determined the decay constant \(k\) by observing that the substance lost 3% of its mass in 6 hours. This was mathematically expressed using natural logarithms to solve for \(k\). The steps included:

• Recognizing the relationship \(e^{-6k} = \frac{97}{100}\).
• Applying the natural logarithm to both sides to isolate \(k\): \(-6k = \ln\left(\frac{97}{100}\right)\).
• Solving for \(k\) by dividing by \(-6\): \(k = -\frac{\ln(\frac{97}{100})}{6}\).

This constant \(k\) allows us to project the remaining substance amount at any point in the future with accuracy.

Natural Logarithm

The natural logarithm is an essential tool in understanding exponential growth and decay. It’s denoted as \(\ln\) and is based on the constant \(e\), which is approximately 2.71828. Natural logarithms are particularly effective when dealing with exponential functions, such as the decay of radioactive materials.In our problem, the natural logarithm helped simplify the equation to find the decay constant \(k\). By taking the natural logarithm of both sides of the equation, we isolated \(k\):

• This is seen when applying \(\ln\) to both sides of the equation \(e^{-6k} = \frac{97}{100}\).
• Doing this makes it easier to manipulate and solve for unknowns in exponential functions.

The power of using \(\ln\) lies in its ability to transform complex exponential relationships into manageable algebraic ones. This allows us to accurately calculate decay rates and predict future states of a decaying substance.