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Let's break down Problem 4 step by step to understand and solve each part.

### Problem 4 Overview:
- **Scenario:** Kerosene must be purified by passing through a pipe containing clay, which removes 20% of pollutants per foot of pipe.
- **Variables:**
  - \( P_0 \) is the initial amount of pollutants.
  - \( P(n) \) represents the amount of pollutants left after \( n \) feet of the pipe.

---

### Part (a): Deriving an Expression for \( P(n) \)

- Every foot of the pipe removes 20% of the pollutants, meaning 80% of the pollutants remain after each foot.
- The relationship is exponential, as each foot leaves 80% of the previous amount of pollutants.

Thus, after \( n \) feet of pipe, the remaining pollutants can be written as:
\[
P(n) = P_0 \times (0.8)^n
\]
Given that \( P_0 = 1 \), the expression becomes:
\[
P(n) = (0.8)^n
\]

---

### Part (b): Graph the Function and Explain Behavior

The function \( P(n) = (0.8)^n \) is an exponential decay function. As \( n \), the length of the pipe, increases, the amount of pollutants decreases rapidly at first, then levels off toward zero but never quite reaches it.

- **Behavior:** The more the length of the pipe increases, the less pollutants remain. However, the amount of pollutants decreases at a diminishing rate as \( n \) grows larger, approaching zero asymptotically but never completely reaching zero.

---

### Part (c): Determine the Length of Pipe to Eliminate 50% of Pollutants

We are given \( P_0 = 1 \), and we want to find \( n \) such that 50% of the pollutants are removed, i.e., \( P(n) = 0.5 \).

Using the equation \( P(n) = (0.8)^n \), set:
\[
0.8^n = 0.5
\]
To solve for \( n \), take the natural logarithm of both sides:
\[
\ln(0.8^n) = \ln(0.5)
\]
\[
n \cdot \ln(0.8) = \ln(0.5)
\]
\[
n = \frac{\ln(0.5)}{\ln(0.8)}
\]
Now, calculating this:

Let me compute \( n \).The length of the pipe required to remove 50% of the pollutants is approximately \( n = 3.11 \) feet.

---

### Recap:

1. **Expression for \( P(n) \):** \( P(n) = (0.8)^n \)
2. **Graph:** The graph of \( P(n) = (0.8)^n \) shows exponential decay, with pollutants approaching zero as the pipe length increases.
3. **50% Elimination:** The pipe needs to be approximately 3.11 feet long to ensure 50% of the pollutants are removed.

---

Would you like more details or have any other questions?

Here are 5 related questions you might find interesting:
1. What happens if the removal rate per foot changes from 20% to 30%?
2. How would the expression \( P(n) \) change if multiple pipes were used in parallel?
3. How long would the pipe need to be to remove 90% of the pollutants?
4. What effect would doubling the initial pollutant concentration have on the pipe length needed to remove 50% of pollutants?
5. How could you modify the equation to account for inefficiencies in pollutant removal over time?

**Tip:** When dealing with exponential decay, always consider the rate of reduction and how it compounds over repeated steps (e.g., feet of pipe).

Before kerosene can be used as jet fuel, federal regulations require that the pollutants in it be removed by passing the kerosene through clay. Suppose that the clay is in a pipe and that each foot of the pipe will remove 20% of the pollutants that enter it.

This problem explores exponential decay in pollutant removal from kerosene as it passes through a clay pipe, where each foot of pipe removes 20% of pollutants. Learn how to derive the decay function and calculate the pipe length needed to remove 50% of pollutants.

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