HALF-LIFE Iodine-131, a radioactive isotope commonly used to treat
thyroid cancer, has a half-life of 8.02 days.
Part A
Write the continuous exponential decay equation for Iodine-131. Round
the value of k to the nearest thousandth.
Part B
How much of a 15-gram sample will be left after 20 days? Round to the
nearest tenth if necessary.
grams

### Part A: Continuous Exponential Decay Equation

The continuous exponential decay equation is of the form:

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

where:
- \( A(t) \) is the amount of substance left after time \( t \),
- \( A_0 \) is the initial amount of the substance,
- \( k \) is the decay constant (rate of decay),
- \( t \) is the time in days,
- \( e \) is Euler's number (approximately 2.71828).

To find the decay constant \( k \), we use the formula:

\[
k = \frac{\ln(2)}{\text{half-life}}
\]

Given the half-life of Iodine-131 is 8.02 days, we calculate \( k \):

\[
k = \frac{\ln(2)}{8.02}
\]

Let me calculate \( k \) for you.The decay constant \( k \) is approximately \( 0.086 \).

Thus, the continuous exponential decay equation for Iodine-131 is:

\[
A(t) = A_0 \cdot e^{-0.086t}
\]

### Part B: How much of a 15-gram sample will be left after 20 days?

We now use the equation \( A(t) = A_0 \cdot e^{-kt} \) to find how much of the 15-gram sample remains after 20 days.

Given:
- \( A_0 = 15 \) grams,
- \( t = 20 \) days,
- \( k = 0.086 \),

We calculate the remaining amount \( A(20) \). Let me compute that.After 20 days, approximately **2.7 grams** of the 15-gram sample of Iodine-131 will remain.

Would you like more details on the process or have any other questions?

Here are 5 related questions to explore further:
1. What is the decay equation for a substance with a different half-life?
2. How does the half-life of a substance affect the decay constant \(k\)?
3. How can you determine the amount of a substance left after multiple half-lives?
4. What is the difference between continuous and discrete decay models?
5. How do we calculate the time required for a sample to decay to a specific amount?

**Tip**: The exponential decay formula is powerful because it can model not only radioactive decay but also processes like population decline and heat dissipation.

HALF-LIFE Iodine-131, a radioactive isotope commonly used to treat thyroid cancer, has a half-life of 8.02 days. Part A: Write the continuous exponential decay equation for Iodine-131. Round the value of k to the nearest thousandth. Part B: How much of a 15-gram sample will be left after 20 days? Round to the nearest tenth if necessary.

Solve the exponential decay equation for Iodine-131 with a half-life of 8.02 days. Learn how to calculate the decay constant and determine how much of a 15-gram sample remains after 20 days. This problem covers key concepts in radioactive decay, exponential functions, and half-life calculations. Suitable for Grades 10-12.

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