Dr. O'Keefe begins an experiment with  units of a radioactive material in his tank. Every day, half of the material in the tank will decompose. How many units of the material will he have left at the end of the th day? Express your answer as a common fraction.

Let's denote the initial amount of radioactive material as \( A_0 \). The problem states that every day, half of the material decomposes, meaning that after each day, the amount of material left is halved.

The amount of radioactive material left after \( t \) days can be expressed using the formula:
\[
A_t = A_0 \times \left(\frac{1}{2}\right)^t
\]
where:
- \( A_0 \) is the initial amount of radioactive material,
- \( A_t \) is the amount of material left after \( t \) days,
- \( t \) is the number of days that have passed.

Since the problem doesn't specify the initial amount \( A_0 \), we'll assume it to be \( A_0 = 1 \) unit for simplicity.

Thus, the amount of material left after \( t \) days is:
\[
A_t = 1 \times \left(\frac{1}{2}\right)^t = \left(\frac{1}{2}\right)^t
\]
So, at the end of the \( t \)th day, the number of units of material left is \( \left(\frac{1}{2}\right)^t \).

For example:
- At the end of the 1st day (\( t = 1 \)), \( A_1 = \left(\frac{1}{2}\right)^1 = \frac{1}{2} \) units.
- At the end of the 2nd day (\( t = 2 \)), \( A_2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \) units.

The general answer is \( \left(\frac{1}{2}\right)^t \) units.

Would you like more details on this solution, or do you have any other questions?

### Follow-up questions:
1. What happens to the material after 10 days?
2. How does the amount of material change between two consecutive days?
3. If the initial amount was 2 units, how would the formula change?
4. How can we model this situation using logarithms?
5. What would be the material left after 0 days?
6. How would the formula change if 40% of the material decomposed daily?
7. How many days would it take for the material to reduce to one-eighth of its original amount?
8. How is this problem related to exponential decay in physics?

**Tip:** In problems involving halving or doubling, the concept of exponential functions often comes into play.

Understand how radioactive material decays over time in Dr. O'Keefe's experiment with a detailed explanation. Learn about exponential decay and the formula A_t = A_0 * (1/2)^t. Suitable for high school students studying exponential functions and decay.

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