The half-life of a radioactive substance is 1200 years. If the initial amount of the substance is 300 grams, give an exponential model for the amount remaining after t years. What amount of substance remains after 1000 years?

To model the amount of radioactive substance remaining after \( t \) years, we use the exponential decay formula:

\[
A(t) = A_0 \cdot e^{kt}
\]

Where:
- \( A(t) \) is the amount remaining after \( t \) years.
- \( A_0 \) is the initial amount of the substance (300 grams in this case).
- \( k \) is the decay constant.
- \( t \) is the time in years.

### Step 1: Calculate the decay constant \( k \)
The half-life of the substance is 1200 years, meaning after 1200 years, half of the substance remains. This gives us the equation:

\[
\frac{1}{2} = e^{k \cdot 1200}
\]

Taking the natural logarithm of both sides:

\[
\ln\left(\frac{1}{2}\right) = k \cdot 1200
\]

\[
k = \frac{\ln\left(\frac{1}{2}\right)}{1200} = \frac{-\ln(2)}{1200}
\]

\[
k \approx \frac{-0.6931}{1200} \approx -0.0005776
\]

### Step 2: Exponential decay model
Now that we have \( k \), we can write the exponential model for the amount remaining after \( t \) years:

\[
A(t) = 300 \cdot e^{-0.0005776t}
\]

### Step 3: Calculate the amount remaining after 1000 years
Substitute \( t = 1000 \) into the model:

\[
A(1000) = 300 \cdot e^{-0.0005776 \cdot 1000}
\]

Let's calculate this:

\[
A(1000) = 300 \cdot e^{-0.5776}
\]

\[
A(1000) \approx 300 \cdot 0.561
\]

\[
A(1000) \approx 168.3 \, \text{grams}
\]

So, approximately 168.3 grams of the substance remains after 1000 years.

Would you like a more detailed breakdown or further clarification?

Here are 5 related questions:
1. How does the half-life affect the decay constant \( k \)?
2. What happens to the decay model if the initial amount \( A_0 \) changes?
3. How would the model change for a substance with a shorter half-life?
4. How is the exponential model used to predict remaining quantities in real-world applications?
5. What are some methods for experimentally determining the half-life of a substance?

**Tip:** Exponential decay models are used in various fields beyond radioactivity, including pharmacology and population studies.

Explore how to model the decay of a radioactive substance with a half-life of 1200 years. Starting with 300 grams, find the remaining amount after 1000 years using the exponential decay formula. This detailed breakdown is ideal for high school students in Grades 10-12.

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