Uranium-238, the most common uranium isotope, has a half-life of 4.5 billion years. After 10,000 years, what percentage of the original amount of uranium remains? This series of problems will build your fluency with exponential growth and decay. For each question below, write a 

function modeling the scenario, and then solve the problem.

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.

Explore the decay of Uranium-238 over 10,000 years, using the exponential decay model. Calculate the percentage of original uranium remaining and understand the relationship between half-life and decay constant. Suitable for high school students studying exponential growth and decay.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation