The half-life of bismuth-210, 210Bi, is 5 days. Estimate the amount remaining after 4 weeks. (Round your answer to one decimal place.)
 
 mg

To estimate the amount of bismuth-210 remaining after 4 weeks, we can use the formula for exponential decay, which is given by:

\[
N(t) = N_0 \times \left( \frac{1}{2} \right)^{\frac{t}{T}}
\]

where:
- \( N(t) \) is the amount remaining after time \( t \),
- \( N_0 \) is the initial amount,
- \( T \) is the half-life,
- \( t \) is the time elapsed.

### Given:
- The half-life \( T \) of bismuth-210 is 5 days.
- The time elapsed \( t \) is 4 weeks = \( 4 \times 7 = 28 \) days.

Now, we can substitute these values into the formula:

\[
N(28) = N_0 \times \left( \frac{1}{2} \right)^{\frac{28}{5}}
\]

Let's calculate the fraction remaining after 28 days, and then round to one decimal place.After 4 weeks (28 days), approximately 2.1% of the initial amount of bismuth-210 will remain. 

If you started with 1 mg, then about 0.021 mg (or 0.02 mg when rounded to two decimal places) would remain.

Would you like further details or have any questions?

Here are some follow-up questions to consider:

1. What is the formula for exponential decay?
2. How does the half-life of a substance affect its decay rate?
3. How do we calculate the remaining quantity of a substance after multiple half-lives?
4. What are other real-world examples of exponential decay?
5. How do we apply exponential decay to other fields, like finance or population growth?

**Tip**: Always keep units consistent when calculating decay, such as converting weeks to days.

The half-life of bismuth-210, 210Bi, is 5 days. Estimate the amount remaining after 4 weeks. (Round your answer to one decimal place.)

Learn how to estimate the remaining amount of bismuth-210 after 4 weeks using exponential decay. This problem covers essential concepts of half-life calculations and the exponential decay formula, suitable for students in Grades 9-12.

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