The question asks to write a function describing the relationship between variables $p$ and $q$, where it states that $p$ varies directly with the square of $q$.

Step-by-step solution:

1. General form of direct variation with the square: Since $p$ varies directly with the square of $q$, the general relationship is: $p = kq^2$ where $k$ is the constant of proportionality.

2. Use the given values to find $k$: The problem gives us that when $q = 7$, $p = 441$. Substituting these values into the equation: $441 = k(7^2)$ $441 = k(49)$ Solving for $k$: $k = \frac{441}{49} = 9$

3. Write the final equation: Now that we know $k = 9$, the function relating $p$ and $q$ is: $p = 9q^2$

4. Check with the provided input $q = 63$: The question shows an input where $q = 63$. To find $p$, substitute $q = 63$ into the equation: $p = 9(63^2)$ First, calculate $63^2$: $63^2 = 3969$ Then: $p = 9 \times 3969 = 35721$ So, $p = 35721$ when $q = 63$.

If you'd like more details or have any further questions, feel free to ask!

Here are 5 related questions:

1. What does "direct variation" mean in mathematical terms?
2. How do you calculate the square of a number, and what are some shortcuts?
3. If $p$ varied inversely with the square of $q$, how would the formula change?
4. What are some real-life examples of variables that vary directly with the square of another variable?
5. How would you solve this problem if the variation was with the cube of $q$?

Tip: When solving problems involving direct variation, always look for the constant of proportionality by using the values provided.