A tumor is injected with 0.6 grams of Iodine-125, which has a decay rate of 1.15% per day. Write an exponential model representing the amount of Iodine-125 reamaining in the tumor after t days. Then user the formula to fin the amount of Ipodine-125 that would remain in the rumor after 60 days.

give exact answer

We can model the amount of Iodine-125 remaining after \( t \) days using an exponential decay formula:

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

Where:
- \( A(t) \) is the amount of Iodine-125 remaining after \( t \) days.
- \( A_0 \) is the initial amount of Iodine-125 (in this case, 0.6 grams).
- \( k \) is the decay constant, which can be determined from the given decay rate of 1.15% per day.
- \( t \) is the time in days.

### Step 1: Determine the decay constant \( k \)

The decay rate is 1.15% per day, which means that 1.15% of the Iodine-125 decays each day. This is represented as:

\[
k = \frac{1.15}{100} = 0.0115
\]

### Step 2: Write the exponential decay model

Now, we substitute the known values into the exponential decay formula:

\[
A(t) = 0.6 \cdot e^{-0.0115t}
\]

This is the model representing the amount of Iodine-125 remaining after \( t \) days.

### Step 3: Calculate the amount of Iodine-125 remaining after 60 days

Substitute \( t = 60 \) into the equation:

\[
A(60) = 0.6 \cdot e^{-0.0115 \cdot 60}
\]

Let's calculate this value.

\[
A(60) = 0.6 \cdot e^{-0.69}
\]

\[
A(60) \approx 0.6 \cdot 0.5012 \approx 0.3007
\]

So, after 60 days, approximately **0.3007 grams** of Iodine-125 would remain in the tumor.

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Do you want further clarification or details? 

Here are 5 questions that expand on the concept:
1. How is the decay constant related to the half-life of a substance?
2. Can you model other elements with different decay rates using the same approach?
3. How would the model change if the decay rate was per hour instead of per day?
4. What is the significance of the decay constant \( k \) in real-world scenarios?
5. How does exponential decay compare to linear decay?

**Tip**: The base of natural logarithms \( e \) is crucial in modeling continuous growth or decay processes, making exponential models widely applicable in various fields.

A tumor is injected with 0.6 grams of Iodine-125, which has a decay rate of 1.15% per day. Write an exponential model representing the amount of Iodine-125 remaining in the tumor after t days. Then use the formula to find the amount of Iodine-125 that would remain in the tumor after 60 days.

Explore how to model the decay of Iodine-125 in a tumor using an exponential decay formula. This example problem includes the decay rate of 1.15% per day and shows how to calculate the remaining amount after 60 days. Suitable for Grades 10-12 students studying exponential functions and decay.

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