To solve this problem, let's first express the relationship given:

If $p$ is inversely proportional to the square of $q$, then we can write this relationship as:

$p = \frac{k}{q^2}$

where $k$ is a constant.

Step 1: Find the constant $k$

We are given that $p = 28$ when $q = 8$. Substituting these values into the equation:

$28 = \frac{k}{8^2}$ $28 = \frac{k}{64}$ $k = 28 \times 64 = 1792$

Step 2: Find $p$ when $q = 4$

Now that we know $k = 1792$, we can use this to find $p$ when $q = 4$. Using the equation $p = \frac{k}{q^2}$:

$p = \frac{1792}{4^2}$ $p = \frac{1792}{16}$ $p = 112$

So, when $q = 4$, $p = 112$.

Final Answer:

$p = 112$

Would you like more details or further clarification? Here are five related questions to extend your understanding:

1. What is the significance of inverse proportionality in real-world applications?
2. How does changing the exponent in the inverse proportionality equation affect $p$?
3. Can you solve for $q$ if you are given $p$ and $k$?
4. What would happen if $p$ were directly proportional to the square of $q$?
5. How does the constant $k$ relate to the initial conditions of a problem?

Tip: In inverse proportionality problems, always check whether the relationship involves powers (like squares or cubes) for precision.