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.