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### Problem 5

Given:
- Initial volume of solution = 80 gallons
- Initial concentration of nitrogen = 25% (0.25)
- Solution added: 8% nitrogen at 10 gallons per minute

To find a function \( C(t) \) giving the concentration of nitrogen \( t \) minutes after the 8% solution starts being added:

1. **Initial nitrogen in the tank:**
\[ \text{Nitrogen in tank} = 0.25 \times 80 = 20 \text{ gallons} \]

2. **Volume of solution at time \( t \):**
\[ V(t) = 80 + 10t \]

3. **Nitrogen added per minute:**
\[ \text{Nitrogen added per minute} = 0.08 \times 10 = 0.8 \text{ gallons} \]

4. **Total nitrogen at time \( t \):**
\[ \text{Total nitrogen} = 20 + 0.8t \]

5. **Concentration function \( C(t) \):**
\[ C(t) = \frac{\text{Total nitrogen}}{\text{Total volume}} = \frac{20 + 0.8t}{80 + 10t} \]

### Problem 18

Given:
\[ C(t) = \frac{200 + 25t}{50 + t} \]

#### (a) Find the initial concentration of salt.
Initial concentration means when \( t = 0 \):
\[ C(0) = \frac{200 + 25 \cdot 0}{50 + 0} = \frac{200}{50} = 4\% \]

#### (b) Find how long it takes for the concentration to reach 10%.
Set \( C(t) = 10 \):
\[ 10 = \frac{200 + 25t}{50 + t} \]
Solving for \( t \):
\[ 10(50 + t) = 200 + 25t \]
\[ 500 + 10t = 200 + 25t \]
\[ 500 - 200 = 25t - 10t \]
\[ 300 = 15t \]
\[ t = \frac{300}{15} = 20 \]

#### (c) Find how long it takes for the concentration to reach 15%.
Set \( C(t) = 15 \):
\[ 15 = \frac{200 + 25t}{50 + t} \]
Solving for \( t \):
\[ 15(50 + t) = 200 + 25t \]
\[ 750 + 15t = 200 + 25t \]
\[ 750 - 200 = 25t - 15t \]
\[ 550 = 10t \]
\[ t = \frac{550}{10} = 55 \]

#### (d) Find how long it takes for the concentration to reach 20%.
Set \( C(t) = 20 \):
\[ 20 = \frac{200 + 25t}{50 + t} \]
Solving for \( t \):
\[ 20(50 + t) = 200 + 25t \]
\[ 1000 + 20t = 200 + 25t \]
\[ 1000 - 200 = 25t - 20t \]
\[ 800 = 5t \]
\[ t = \frac{800}{5} = 160 \]

#### (e) What is the concentration approaching as \( t \to \infty \)?
To find the limit as \( t \to \infty \):
\[ \lim_{t \to \infty} C(t) = \lim_{t \to \infty} \frac{200 + 25t}{50 + t} = \lim_{t \to \infty} \frac{25t}{t} = 25 \]

### Summary

1. **Problem 5**: \( C(t) = \frac{20 + 0.8t}{80 + 10t} \)
2. **Problem 18a**: Initial concentration is 4%
3. **Problem 18b**: Time to reach 10% is 20 minutes
4. **Problem 18c**: Time to reach 15% is 55 minutes
5. **Problem 18d**: Time to reach 20% is 160 minutes
6. **Problem 18e**: As \( t \to \infty \), \( C(t) \) approaches 25%

Do you want more details or have any questions? 

Here are five questions you might want to ask:
1. How do you solve a differential equation for concentration problems?
2. What are the steps to find the limit of a function as \( t \) approaches infinity?
3. Can you explain how to derive the formula for \( C(t) \) in Problem 5?
4. What are other real-life applications of concentration functions?
5. How can these concentration problems be solved using computational tools?

**Tip:** When solving for the time to reach a certain concentration, always carefully isolate the variable and simplify step by step to avoid errors.

Learn how to solve concentration problems involving nitrogen and salt with detailed step-by-step solutions. This includes finding initial concentrations, time to reach specific concentrations, and the limit of concentrations as time approaches infinity. Suitable for advanced high school and college students.

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