f pp is inversely proportional to the square of qq, and pp is 28 when qq is 8, determine pp when qq is equal to 4.

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.

If p is inversely proportional to the square of q, and p is 28 when q is 8, determine p when q is equal to 4.

Learn how to solve an inverse proportionality problem where p is inversely proportional to the square of q. Given p = 28 when q = 8, this step-by-step solution finds p when q = 4. Suitable for students in Grades 8-10.

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