To solve this problem, we will model the decay of Uranium-238 using an exponential decay function. Here's how we can approach the problem:

Exponential Decay Function

The general formula for exponential decay is:

$N(t) = N_0 \cdot e^{-\lambda t}$

Where:

• $N(t)$ is the amount of the substance remaining after time $t$.
• $N_0$ is the initial amount of the substance.
• $\lambda$ is the decay constant.
• $t$ is the time elapsed.

Step 1: Calculate the Decay Constant $\lambda$

The decay constant $\lambda$ is related to the half-life $T_{1/2}$ by the following formula:

$\lambda = \frac{\ln(2)}{T_{1/2}}$

Given that the half-life $T_{1/2}$ of Uranium-238 is 4.5 billion years, we can calculate $\lambda$:

$\lambda = \frac{\ln(2)}{4.5 \times 10^9 \text{ years}}$

Step 2: Model the Scenario

Now we want to find the percentage of the original amount of uranium remaining after 10,000 years. The function modeling the scenario is:

$N(10000) = N_0 \cdot e^{-\lambda \cdot 10000}$

To find the percentage of the original amount remaining, we divide $N(10000)$ by the initial amount $N_0$ and multiply by 100:

$\text{Percentage remaining} = \left( \frac{N(10000)}{N_0} \right) \times 100 = e^{-\lambda \cdot 10000} \times 100$

Step 3: Compute the Result

Now, let's calculate the exact percentage:

1. Calculate the decay constant $\lambda$: $\lambda = \frac{\ln(2)}{4.5 \times 10^9}$

2. Substitute $\lambda$ and $t = 10000$ years into the percentage formula: $\text{Percentage remaining} = e^{-\lambda \cdot 10000} \times 100$

Let's compute this value.After 10,000 years, approximately 99.9998% of the original amount of Uranium-238 remains.

Would you like more details or have any other questions? Here are five related questions:

1. What is the percentage of Uranium-238 remaining after 1 million years?
2. How long will it take for only 50% of Uranium-238 to remain?
3. How would the remaining percentage change if the half-life were shorter?
4. What is the remaining percentage after three half-lives?
5. How does the decay of Uranium-238 compare to that of other radioactive isotopes?

Tip: Understanding the relationship between the half-life and the decay constant is crucial for modeling and predicting decay processes accurately.