Problem 18 - Calculus: Early Transcendentals

Question

A radioactive element decays with a half-life of 6 years. If a mass of the element weighs ten pounds at \(t=0,\) find the amount of the element at time \(t\).

Step-by-Step Solution

Answer

The amount at time \(t\) is given by \(N(t) = 10 \times \left(\frac{1}{2}\right)^{\frac{t}{6}}\).

Step 1: Define the Decay Function

The decay of a radioactive substance can be modeled by the exponential decay function:\[ N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]where \(N(t)\) is the amount of substance at time \(t\), \(N_0\) is the initial amount, and \(T_{1/2}\) is the half-life of the substance.

Step 2: Identify Parameters

From the problem, we know the initial mass \(N_0 = 10\) pounds and the half-life \(T_{1/2} = 6\) years. We will use these parameters in the decay function.

Step 3: Substitute Parameters into Decay Function

Substituting the known values into the decay formula gives:\[ N(t) = 10 \times \left(\frac{1}{2}\right)^{\frac{t}{6}} \]

Step 4: Solve for Various Times

To find the amount of the element at a specific time \(t\), simply substitute the value of \(t\) into the equation from Step 3. For example:- At \(t = 0\): \[ N(0) = 10 \times \left(\frac{1}{2}\right)^{0} = 10 \text{ pounds} \]- At \(t = 6\): \[ N(6) = 10 \times \left(\frac{1}{2}\right)^{1} = 5 \text{ pounds} \]- At \(t = 12\): \[ N(12) = 10 \times \left(\frac{1}{2}\right)^{2} = 2.5 \text{ pounds} \]

Key Concepts

Exponential Decay Function

Exponential decay functions describe the decrease of a quantity at a rate proportional to its current value. In the context of radioactive decay, it means that as time passes, a radioactive substance decreases in quantity, but not in a linear fashion. Instead, the rate of decay is exponential, making it more rapid initially and then slowing down over time.

The general formula for an exponential decay function is:

• \[ N(t) = N_0 imes e^{-kt} \]

where:

• \( N(t) \) is the amount of the substance at time \( t \),
• \( N_0 \) is the initial amount,
• \( k \) is the decay constant.

In radioactive decay, this formula can be further expressed in terms of half-life, simplifying how we understand changes over time. Understanding this growth model is crucial for predicting how much of a substance remains after a certain period.

Half-life

Half-life is the time required for half of the radioactive substance to decay or transform into another element or isotope. It is a critical property in understanding the behavior of radioactive materials.

The half-life is denoted as \( T_{1/2} \) and it is used not only in physics, but also in other disciplines such as archaeology and medicine, to determine timelines and dosages. For our purposes, we see this concept represented in the radioactive decay formula:

• \[ N(t) = N_0 \times \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \]

By using the half-life in this function, we can quickly solve for \( N(t) \), the remaining quantity of the substance at any given time \( t \).

For example, if a substance has a half-life of 6 years, every 6 years, the amount of the substance would be half of its current amount. This notion simplifies calculations and helps in visualizing the decay process.

Radioactivity in Calculus

Radioactivity problems are a prime example of how calculus can be applied to real-world situations. Radioactive decay is modeled using calculus concepts to predict the future amounts of substances.

The decay process is continuous, and calculus provides us the tools to deal with such continuous phenomena. A key component here is the decay constant \( k \), which relates directly to the half-life through the formula:

• \[ k = \frac{\ln(2)}{T_{1/2}} \]

Once the decay constant is known, we can fully integrate it into the exponential decay model. Calculus allows us to understand not just the rate of change at any instant, but also how much a quantity will change over time. This is particularly important for industries that use radioactive elements, as they need precise calculations for safety and effectiveness. By using integration and differentiation, fields like nuclear medicine and archaeological dating become more exact and reliable.