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The problem presented involves radioactive decay, specifically for Uranium-240. The function given is:

\[
A(t) = 4700 \left( \frac{1}{2} \right)^{\frac{t}{14}}
\]

Where:
- \( A(t) \) represents the amount of Uranium-240 left after \( t \) hours.
- The initial amount of Uranium-240 is the value of \( A(0) \).
- We are asked to find the amount after 50 hours, i.e., \( A(50) \).

### Step 1: Finding the initial amount
The initial amount corresponds to \( t = 0 \):

\[
A(0) = 4700 \left( \frac{1}{2} \right)^{\frac{0}{14}} = 4700 \times \left( \frac{1}{2} \right)^0 = 4700 \times 1 = 4700 \text{ grams}
\]

### Step 2: Finding the amount after 50 hours
We substitute \( t = 50 \) into the formula:

\[
A(50) = 4700 \left( \frac{1}{2} \right)^{\frac{50}{14}} = 4700 \times \left( \frac{1}{2} \right)^{3.571}
\]

Now I'll compute the value of \( A(50) \).The amount of Uranium-240 remaining after 50 hours is approximately 395.36 grams.

### Summary:
- **Initial amount**: 4700 grams
- **Amount after 50 hours**: 395.36 grams (rounded to two decimal places)

Would you like further details or clarification? Here are 5 related questions for exploration:

1. How would the remaining amount change after 100 hours?
2. What is the percentage of Uranium-240 remaining after 50 hours?
3. How is the half-life concept applied in other radioactive materials?
4. What is the general form of the exponential decay function?
5. How long would it take for only 10% of the Uranium-240 to remain?

**Tip**: The half-life formula is widely used in fields like chemistry, biology, and even finance (for depreciation and other time-based decay models).

The radioactive substance uranium-240 has a half-life of 14 hours. The amount A(t) of a sample of uranium-240 remaining (in grams) after t hours is given by the exponential function: A(t) = 4700 (1/2)^(t/14). Find the initial amount in the sample and the amount remaining after 50 hours.

Solve for the initial amount and the remaining amount of uranium-240 after 50 hours using exponential decay. Learn how to apply the half-life formula A(t) = 4700(1/2)^(t/14) with step-by-step guidance. This problem is ideal for students in high school, covering concepts of radioactive decay and half-life calculations.

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