A large brine tank containing a solution of salt and water is being diluted with fresh water.
The relationship between the elapsed time,
\[t\], in hours, after the dilution begins, and the concentration of salt in the tank,
\[S(t)\], in grams per liter
\[(\text{g/l})\], is modeled by the following function.
\[ S(t)=500\cdot e^{{-0.25t}}\]
What will the concentration of salt be after
\[10\] hours?
Round your answer, if necessary, to the nearest hundredth.

To find the concentration of salt in the brine tank after 10 hours, we need to evaluate the given function:

\[
S(t) = 500 \cdot e^{-0.25t}
\]

at \( t = 10 \). This will provide us with the concentration of salt in grams per liter at that specific time.

### Step-by-Step Calculation:

1. **Substitute \( t = 10 \) into the equation**:
   \[
   S(10) = 500 \cdot e^{-0.25 \cdot 10}
   \]

2. **Calculate the exponent**:
   \[
   -0.25 \cdot 10 = -2.5
   \]

3. **Evaluate the expression \( e^{-2.5} \)**:
   - The value of \( e^{-2.5} \) can be computed using a calculator. The constant \( e \) (Euler's number) is approximately \( 2.71828 \).
   \[
   e^{-2.5} \approx 0.08208
   \]

4. **Calculate \( S(10) \)**:
   \[
   S(10) = 500 \cdot 0.08208
   \]

5. **Multiply to find the concentration**:
   \[
   S(10) = 41.04
   \]

### Final Answer:
The concentration of salt after 10 hours is approximately **41.04 grams per liter (g/l)**.

Thus, the concentration of salt after 10 hours is \( 41.04 \, \text{g/l} \).

---

If you have any questions or need further explanation, feel free to ask. 

### Relative Questions:
1. What is the rate of change of the concentration at \( t = 10 \) hours?
2. How does the concentration change between \( t = 5 \) and \( t = 10 \) hours?
3. If the initial concentration was \( 500 \, \text{g/l} \), how much does it decrease after 10 hours?
4. How can we determine when the concentration reaches \( 25 \, \text{g/l} \)?
5. What happens to the concentration as \( t \) approaches infinity?
6. How long will it take for the concentration to be halved?
7. Can you derive the function's decay constant from the equation?
8. How would adding salt affect the model equation?

### Tip:
When dealing with exponential decay problems, remember that the base of the natural logarithm \( e \) is approximately 2.71828, and it plays a crucial role in calculating growth or decay rates.

Learn how to calculate the concentration of salt in a brine tank after 10 hours using the exponential decay model. This problem involves understanding exponential functions and their application in real-world scenarios. Suitable for advanced high school students.

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