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To solve this problem, let’s go through each part carefully.

### Given Information:
1. \( p \) varies directly with \( q^3 \). This can be written as:
   \[
   p = k q^3
   \]
   where \( k \) is the constant of proportionality.

2. When \( q = 3 \), \( p = 81 \).

### Steps to Solve:

#### 1. **Finding the Constant of Proportionality \( k \):**
   Substitute \( p = 81 \) and \( q = 3 \) into the equation \( p = k q^3 \):
   \[
   81 = k \cdot (3)^3
   \]
   \[
   81 = k \cdot 27
   \]
   Solving for \( k \):
   \[
   k = \frac{81}{27} = 3
   \]

   So, the constant of proportionality \( k \) is **3**.

#### 2. **Writing the Function that Relates \( p \) and \( q \):**
   Now that we know \( k = 3 \), we can write the function as:
   \[
   p = 3 q^3
   \]

#### 3. **Finding \( p \) when \( q = 2 \):**
   Substitute \( q = 2 \) into the function \( p = 3 q^3 \):
   \[
   p = 3 \cdot (2)^3
   \]
   \[
   p = 3 \cdot 8
   \]
   \[
   p = 24
   \]

   So, when \( q = 2 \), \( p \) is **24**.

### Summary of Answers:
- **Constant of proportionality, \( k = 3 \)**
- **Function relating \( p \) and \( q \): \( p = 3 q^3 \)**
- **If \( q = 2 \), then \( p = 24 \)**

Would you like further details on any step? 

Here are some related questions for additional practice:
1. What happens to \( p \) if \( q \) doubles?
2. How would you find \( p \) if \( p \) varied with \( q^2 \) instead of \( q^3 \)?
3. What is the effect on \( p \) if \( k \) is doubled?
4. If \( p = 48 \), what is the value of \( q \)?
5. How would you solve for \( k \) if \( p = 50 \) and \( q = 2 \)?

**Tip:** Always double-check your substitutions when solving for constants in proportional relationships.

p varies directly with q^3 and when q = 3, p = 81. Give the constant of proportionality k, the function that relates p and q, and the value of p when q = 2.

Solve a direct variation problem where p varies with q^3. Find the constant of proportionality k, the function relating p and q, and the value of p when q = 2. This solution includes key concepts like exponents and proportionality, ideal for students in Grades 8-10.

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