Problem 70 - College Algebra and Calculus: An Applied Approach

Question

Radioactive Decay One hundred grams of radium \(\left({ }^{226} \mathrm{Ra}\right)\) is stored in a container. The amount of radium \(R\) present after \(t\) years is given by \(R=100 e^{-0.0004335 t}\). (a) Use a graphing utility to graph this function over the interval from \(t=0\) to \(t=10,000\). (b) How much of the 100 grams of radium will remain after 10,000 years? (c) Use the graph to estimate the half-life of \({ }^{226} \mathrm{Ra}\). Explain your reasoning.

Step-by-Step Solution

Answer

Here are the short answers for each section of the question: (a) Refer to the graphing utility for the visual representation. The curve should show a characteristic exponential decay shape, starting from 100 grams and gradually decreasing to near-zero over time. (b) After substituting \(t = 10,000\) into the equation and performing the necessary calculations, the quantity of radium left can be found. (c) By estimating where 50 grams is reached on the graph (Y-axis), the corresponding time on the X-axis can be identified, which represents the approximate half-life of Radium-226.

Step 1: Graphing the Function

To graph this function over the interval \(t=0\) to \(t=10,000\), a graphing utility can be used. Input the function \(R=100 e^{-0.0004335 t}\) into the utility and set the time interval as specified. This will yield a decay curve, demonstrating how radium decays over time.

Step 2: Calculating remaining Radium after 10,000 years

To determine how much of the 100 grams of radium remains after 10,000 years, substitute \(t = 10,000\) into the equation to get \(R=100 e^{-0.0004335*10000}\). By calculating this, we find the amount of radium remaining after 10,000 years.

Step 3: Estimating Half-Life

The half-life of a substance is the time taken for half of the substance to decay. This means when \(R = 50\) (since our initial quantity is 100 grams), it's at half-life. Estimate this from the graph where \(R\) intersects with 50, which should reflect the half-life of \({ }^{226} \mathrm{Ra}\).

Key Concepts

Exponential Decay

Exponential decay describes the process by which a quantity decreases at a rate proportional to its current value. Specifically, radioactive decay is a classic example of this phenomenon, where unstable atomic nuclei lose energy by emitting radiation. In algebra, the exponential decay function can be expressed as:
\[ R(t) = R_0 e^{-kt} \]
where \( R(t) \) is the amount of substance at time \( t \), \( R_0 \) is the initial amount, \( e \) is the base of the natural logarithm, \( k \) is the decay constant, and \( t \) is the time. The negative sign indicates that the amount decays over time. Understanding the nature of exponential decay is fundamental in subjects like physics, chemistry, biology, and even finance.

Half-Life Calculation

The half-life of a radioactive element is the time required for half of the element to decay. It is a constant characteristic of each radioactive isotope and is crucial in the fields of archaeology, geology, and medicine for dating artifacts, rocks, and analyzing substances.
The half-life can be calculated using the exponential decay formula:
\[ t_{\text{half}} = \frac{\ln(2)}{k} \]
where \( t_{\text{half}} \) represents the half-life and \( k \) is the decay constant. Thus, once \( k \) is known, calculating the half-life becomes a straightforward task. In an educational context, mastering half-life calculations is important for accurately predicting how long a substance remains active or hazardous.

Graphing Exponential Functions

Graphing exponential functions is an essential skill in understanding the behavior of quantities that change continuously over time. An exponential decay graph is typically a curve that slopes downwards, reflecting the decrease in the quantity.
When graphing the function \( R=100 e^{-0.0004335 t} \), we can visualize the rate of decay of radium. For educational purposes, using graphing utilities can help students confirm the theoretical outcomes of calculations. Moreover, observing the steepness of the curve provides insight into how quickly the quantity is decaying, which is particularly helpful in radioactive decay scenarios. Graphing also allows for the visualization of the half-life by identifying the point where the quantity has reduced to half of its initial value.

Calculus Applications

Calculus plays a significant role in understanding and analyzing exponential decay. The derivative of an exponential decay function with respect to time gives the rate of change of the quantity, which is essential in determining how fast a substance is decaying at any point in time.
For example, to analyze the rate of change of radium in the provided exercise, you would take the derivative of the decay function \( R(t) \) with respect to \( t \), which yields:
\[ \frac{dR}{dt} = -0.0004335 \cdot 100 e^{-0.0004335 t} \]
The negative sign indicates that radium is decreasing. In real-world applications, calculus is used to solve complex decay problems, optimize processes in engineering, and aid in environmental predictions related to radioactive materials.