Problem 69 - Precalculus with Limits

Question

Radioactive Decay Let \(Q\) represent a mass of radioactive plutonium \((239 \mathrm{Pu})(\) in grams), whose half-life is \(24,100\) years. The quantity of plutonium present after \(t\) years is \(Q=16\left(\frac{1}{2}\right)^{t / 24,100}\) (a) Determine the initial quantity (when \(t=0 )\) . (b) Determine the quantity present after \(75,000\) years. (c) Use a graphing utility to graph the function over the interval \(t=0\) to \(t=150,000 .\)

Step-by-Step Solution

Answer

The initial quantity of plutonium is 16 grams. After 75,000 years, there will be approximately 1 gram of plutonium left.

Step 1: Calculate the initial quantity

To find the initial quantity of plutonium (i.e., when \(t=0\)), we apply the given formula directly. So: \(Q=16\left(\frac{1}{2}\right)^{0 / 24,100}\). As the exponent here turns out to be 0, we know that any number to the power of 0 is always 1. So, \(Q\) equals 16.

Step 2: Calculate the quantity after 75,000 years

Next, let's find the quantity of plutonium after 75,000 years. By replacing \(t\) with 75,000 in the formula, we get: \(Q=16\left(\frac{1}{2}\right)^{75,000 / 24,100}\). We can compute the exponent first, which gives us: \(Q=16(0.5)^{3.111618}\). Finally, we find \(Q\) by taking the given base to the computed power and multiplying by 16, which gives us 1 gram.

Step 3: Graph the function

To graph the function, a plotting device or utility can be used. By plotting \(Q=16\left(\frac{1}{2}\right)^{t / 24,100}\) on the domain \([0, 150,000]\), a standard decay curve portraying the decrease in plutonium over the years will be displayed. However, this is not feasible to display here, and is left to the user.

Key Concepts

Half-Life

The concept of half-life is crucial in understanding radioactive decay. It refers to the time required for half of a given amount of a radioactive substance to decay. In other words, if you start with a specific quantity of a radioactive material, after one half-life, only half of it will remain. For plutonium-239, as given in our exercise, the half-life is precisely 24,100 years.
This means every 24,100 years, the amount of plutonium-239 present will reduce by half.
Knowing the half-life helps scientists and researchers predict how long a substance will remain radioactive, which is essential in fields like nuclear medicine and environmental science. The mathematical representation used for our problem is \[ Q = 16\left(\frac{1}{2}\right)^{t / 24,100} \] where:

• Q is the quantity of plutonium left after time t.
• 16 is the initial amount.
• The power \(\frac{t}{24,100}\) shows how many half-lives have passed.

Exponential Decay

Exponential decay describes the process of a material reducing at a consistent rate over a period. The decay is 'exponential' because the rate of decay is proportional to the current amount—meaning it decreases more rapidly when there's a lot, and less as time goes on.
In mathematical terms, the decay follows a curve that slopes downward, indicating a rapid loss initially, tapering off over time. For our plutonium-239 example, the formula \[ Q = 16\left(\frac{1}{2}\right)^{t / 24,100} \] reflects exponential decay. The term \(\left(\frac{1}{2}\right)^{t / 24,100}\) indicates that as time t increases, the factors of half continue to multiply, decreasing the amount of plutonium significantly.
Therefore, when you plot this function, the graph represents a smooth and continuous decline, showcasing the decaying nature of radioactive materials.

Graphing Functions

Graphing functions provides a visual representation of mathematical equations. For radioactive decay, it powerfully illustrates how a substance diminishes over time.
In our context, the graph of the function \[ Q = 16\left(\frac{1}{2}\right)^{t / 24,100} \]from t = 0 to 150,000 years will show a classic decay curve.
This curve starts high (at 16 grams when t=0) and gradually slopes downward, approaching zero as time moves onward.To create such a graph, you can use graphing utilities like Desmos or a graphing calculator:

• Input the function into the calculator or software.
• Define the domain for t (e.g., from 0 to 150,000 years).
• Observe how the value of Q decreases as t increases, confirming the half-life period and decay rate.

This visual aid makes it easier to comprehend the impact of time on the amount of plutonium-239 and on radioactive substances generally.

Plutonium-239

Plutonium-239 is a radioactive isotope of plutonium with significant applications in nuclear technology. Due to its ability to sustain a nuclear chain reaction, it's commonly used in nuclear reactors and as a material in nuclear weapons.
This isotope has a half-life of 24,100 years, meaning it decays slowly, presenting both benefits and challenges. From a scientific and environmental perspective, understanding its decay is crucial because of its long-term radiation levels.
When handling plutonium-239, careful considerations must be taken due to its high toxicity and radioactivity:

• Safe storage: It requires secure, long-term storage solutions to prevent contamination.
• Health risks: Prolonged exposure can lead to severe health impacts, reinforcing the need for strict safety protocols.
• Environmental impact: Any release into the environment could have lasting effects, necessitating thorough monitoring and management.

These factors highlight why the calculation of its decay through the concept of half-life is so crucial. It helps predict and mitigate the impacts of this potent substance.