https://file.aiquickdraw.com/mathbot/file/17299838381534e35o5l6.jpg

To solve this problem, let’s analyze the given information:

### 1. **Initial Amount Calculation**  
The function for the amount of uranium-240 remaining after \( t \) hours is:
\[
A(t) = 2800 \left(\frac{1}{2}\right)^{\frac{t}{14}}
\]

The initial amount refers to the amount of uranium-240 present at \( t = 0 \):
\[
A(0) = 2800 \left(\frac{1}{2}\right)^{\frac{0}{14}} = 2800 \times 1 = 2800 \text{ grams}
\]
So, the **initial amount** is **2800 grams**.

### 2. **Amount Remaining After 30 Hours**  
We are asked to find the amount remaining after 30 hours. Substitute \( t = 30 \) into the function:
\[
A(30) = 2800 \left(\frac{1}{2}\right)^{\frac{30}{14}}
\]

Calculate \( \frac{30}{14} \) first:
\[
\frac{30}{14} \approx 2.142857
\]

Now substitute and calculate:
\[
A(30) = 2800 \left(\frac{1}{2}\right)^{2.142857}
\]

Using the exponential function:
\[
A(30) \approx 2800 \times 0.226
\]
\[
A(30) \approx 633 \text{ grams (rounded to the nearest gram)}
\]

### Final Answers
- **Initial amount**: 2800 grams
- **Amount after 30 hours**: 633 grams

Would you like further details or have any questions?

### Related Questions to Expand Your Knowledge
1. How would the function change if the half-life of uranium-240 was different?
2. What would be the amount of uranium-240 remaining after 60 hours?
3. How does changing the initial quantity affect the decay function?
4. What is the meaning of a half-life in radioactive decay, and why does it result in exponential decay?
5. How could you use logarithms to find the time it takes for the sample to decay to a specific amount?

#### Tip:
When solving exponential decay problems, remember that the decay factor \((\frac{1}{2})\) corresponds to the half-life.

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 following exponential function.

A(t) = 2800 (1/2)^(t/14)

Find the initial amount in the sample and the amount remaining after 30 hours. Round your answers to the nearest gram as necessary.

Calculate the initial amount and the amount remaining after 30 hours for a uranium-240 sample with a half-life of 14 hours using the exponential decay formula. This problem covers key topics in exponential decay, half-life, and radioactive decay. Suitable for high school students in Grades 10-12.

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