The​ half-life of radium is 1690 years. If 90 grams are present​ now, how much will be present in 800​years?

The formula for exponential decay is:

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

Where:
- \( A(t) \) is the amount remaining after time \( t \),
- \( A_0 \) is the initial amount (90 grams in this case),
- \( k \) is the decay constant,
- \( t \) is the time that has passed (800 years),
- \( e \) is Euler's number, approximately 2.71828.

### Step 1: Find the decay constant \( k \)
The decay constant \( k \) is related to the half-life \( T_{1/2} \) by the equation:

\[
k = \frac{\ln(2)}{T_{1/2}}
\]

Given the half-life of radium is 1690 years:

\[
k = \frac{\ln(2)}{1690}
\]

### Step 2: Calculate the remaining amount
Once we have \( k \), we can substitute the known values to find \( A(800) \).

Let me calculate this.After 800 years, approximately 64.83 grams of radium will remain from the initial 90 grams.

Do you want further details or have any questions?

Here are 5 related questions to explore:
1. What is the formula for half-life and how is it derived?
2. How does the decay constant \( k \) affect the rate of decay?
3. What happens if you double the time passed (e.g., to 1600 years)?
4. How do you calculate the remaining amount for a different element with a different half-life?
5. What would the initial mass need to be to have 45 grams left after 800 years?

**Tip:** The decay constant \( k \) is key in determining how quickly a substance decays. The larger the value of \( k \), the faster the decay.

The half-life of radium is 1690 years. If 90 grams are present now, how much will be present in 800 years?

Explore how to calculate the remaining amount of radium after 800 years given a half-life of 1690 years. Starting with 90 grams, this step-by-step solution explains the exponential decay formula and how to calculate the decay constant. Ideal for students in Grades 10-12.

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