Problem 70 - Precalculus with Limits

Question

Radioactive Decay Let \(Q\) represent a mass of carbon 14\((14 \mathrm{C})\) (in grams), whose half-life is 5715 years. The quantity of carbon 14 present after \(t\) years is \(Q=10\left(\frac{1}{2}\right)^{t / 5715}\) . (a) Determine the initial quantity (when \(t=0 )\) . (b) Determine the quantity present after 2000 years. (c) Sketch the graph of this function over the interval \(t=0\) to \(t=10,000 .\)

Step-by-Step Solution

Answer

The initial quantity of Carbon-14 is 10 grams. After 2000 years, the remaining quantity can be obtained from the formula \(Q = 10(\frac{1}{2})^{2000/5715}\). The graph of the function over the interval \(t=0\) to \(t=10000\) is an exponential decay curve.

Step 1: Determine the initial quantity

When \(t=0\), we substitute this into the decay function provided to find the initial quantity. So, \(Q = 10(\frac{1}{2})^{0/5715}\). Using the fact that any number raised to the power of 0 is 1, we find \(Q = 10\).

Step 2: Determine the quantity present after 2000 years

Now, let's find the quantity of Carbon-14 that remains after 2000 years. We substitute \(t=2000\) into our equation and calculate: \(Q = 10(\frac{1}{2})^{2000/5715}\). This will give us the quantity remaining after 2000 years.

Step 3: Sketching the graph for the range

The final task is to graph the function over the interval \(t=0\) to \(t=10000\). The function \(Q = 10(\frac{1}{2})^{t/5715}\) is an instance of exponential decay as already mentioned, so it will start at \(Q=10\) when \(t=0\), and decrease towards \(Q=0\) as \(t\) increases towards \(10000\). Remember, this is an exponential decay, so the graph should show a rapidly decreasing curve that becomes less steep as \(Q\) approaches 0.

Key Concepts

Half-Life

The concept of half-life is essential in understanding radioactive decay. It is the time required for a substance, like carbon-14, to reduce by half through a natural process. In other words, after one half-life period, half of the original substance has decayed.
For carbon-14, which is commonly used in radiocarbon dating, the half-life is 5715 years. This means that if you start with 10 grams of carbon-14, after 5715 years, you will have just 5 grams remaining.
Understanding half-life helps us predict how long a sample will remain radioactive. It's a constant rate, unaffected by external factors like temperature or pressure, which makes it very reliable in scientific calculations.

Exponential Decay

Exponential decay describes a process where quantities decrease at a rate proportional to their current value. It's a natural process seen in many domains, including finance and science.
For radioactive substances like carbon-14, the decay follows an exponential curve defined by the equation: \(Q = Q_0 \left( \frac{1}{2} \right)^{t / T}\).

• \(Q_0\) is the initial quantity,
• \(t\) is the time elapsed, and
• \(T\) is the half-life of the substance.

This equation shows how the quantity decreases rapidly at the start and then slows as time goes on.
The shape of the decay curve confirms the rapid initial loss and a slow long-term decline, making the understanding of exponential decay vital for predictions over various time scales.

Carbon-14 Decay

Carbon-14 decay is a specific case of radioactive decay, where carbon-14 isotopes turn into nitrogen-14 through beta decay, a process releasing radiation. This transformation follows the general principles of half-life and exponential decay.
In the context of carbon-14 dating, the decay formula \(Q=10\left(\frac{1}{2}\right)^{t / 5715}\) describes how a sample reduces over time. This is useful for determining the age of archaeological and geological samples by measuring the remaining amount of carbon-14.
As you analyze the decay graph over 10,000 years, expect a steep decline initially, flattening as the sample becomes older. This predictable pattern is why carbon-14 is used to date fossils up to 50,000 years old, providing insights into past eras and extinct species.