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 exponential decay model for the amount of Iodine-125 remaining in the tumor after \( t \) days can be written as:

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

Where:
- \( A(t) \) is the amount of Iodine-125 remaining after \( t \) days.
- \( A_0 \) is the initial amount of Iodine-125, which is \( 0.9 \) grams.
- \( k \) is the decay constant, which can be derived from the decay rate of 1.15% per day (as a negative rate).
- \( t \) is the time in days.

First, let's determine the decay constant \( k \). Since the decay rate is 1.15% per day, we can express it as a decimal:

\[
k = -0.0115
\]

Thus, the exponential decay model becomes:

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

### Now, to find the amount of Iodine-125 remaining after 60 days:

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

Let's compute the exact amount remaining after 60 days.After 60 days, the amount of Iodine-125 remaining in the tumor is approximately \( 0.4514 \) grams.

### Formula Summary:
\[
A(t) = 0.9 \cdot e^{-0.0115t}
\]
And after 60 days:
\[
A(60) \approx 0.4514 \text{ grams}
\]

Would you like further details on the process or any clarifications?

Here are some related questions for you:
1. How would the decay model change if the decay rate were different?
2. How long would it take for the Iodine-125 to decay to half its initial amount?
3. What is the significance of the decay constant \( k \) in this model?
4. How can you determine the decay rate if you know the half-life of a substance?
5. What would the model look like for continuous growth instead of decay?

**Tip:** When solving exponential decay problems, the decay constant \( k \) is always negative, representing a reduction over time.

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.

Learn how to create an exponential decay model to calculate the amount of Iodine-125 remaining after a certain number of days. This problem uses a decay rate of 1.15% per day and covers essential mathematical concepts such as exponential functions and radioactive decay. Find out how much Iodine-125 remains after 60 days.

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