A tumor is injected with  0.9 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.

 

Enter the exact answer.

Enclose arguments of functions in parentheses and include a multiplication sign between terms. For example,  c*exp(h) or  c*ln(h).     A(t) = 

The given problem involves an exponential decay process. The general form of an exponential decay model is:

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

where:
- \( A(t) \) is the amount of substance remaining after \( t \) days,
- \( A_0 \) is the initial amount of the substance (in this case, 0.9 grams),
- \( k \) is the decay constant, and
- \( t \) is the time in days.

### Step 1: Determine the decay constant \( k \)
The problem states that Iodine-125 has a decay rate of 1.15% per day. This is a relative decay rate, meaning that:

\[
k = 0.0115
\]

### Step 2: Write the exponential decay model
Now we substitute the known values into the general model:

\[
A(t) = 0.9 \cdot e^{-0.0115t}
\]

### Step 3: Calculate the amount remaining after 60 days
To find the amount of Iodine-125 remaining after 60 days, substitute \( t = 60 \) into the equation:

\[
A(60) = 0.9 \cdot e^{-0.0115 \cdot 60}
\]

Now let's calculate the exact value.

\[
A(60) = 0.9 \cdot e^{-0.69}
\]

Using a calculator to find the value of \( e^{-0.69} \):

\[
A(60) \approx 0.9 \cdot 0.5012 = 0.4511
\]

Thus, the amount of Iodine-125 remaining after 60 days is approximately 0.4511 grams.

### Final Answer:
\[
A(t) = 0.9 \cdot e^{-0.0115t}
\]

The amount remaining after 60 days is approximately 0.4511 grams.

Would you like more details on this or have any questions? Here are some related questions:

1. How would the model change if the decay rate was 2% per day?
2. What is the half-life of Iodine-125 based on this decay rate?
3. How long would it take for the Iodine-125 to decay to 50% of its original amount?
4. How does the exponential decay model differ from linear decay?
5. What is the meaning of the decay constant \( k \) in real-world applications?

**Tip**: In exponential decay, small changes in the decay rate \( k \) can have a significant impact on long-term outcomes.

A tumor is injected with 0.9 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.

Solve a real-world exponential decay problem involving Iodine-125 with a 1.15% daily decay rate. Learn how to calculate the amount remaining after 60 days using an exponential decay model. Suitable for students in Grades 9-12.

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