https://file.aiquickdraw.com/mathbot/file/1730725244223p26gqsyg.jpg

To solve this problem, let's go through the steps.

### Step 1: Set up the relationship
Since \( y \) is inversely proportional to \( x^2 \), we can express this as:
\[
y = \frac{k}{x^2}
\]
where \( k \) is a constant.

### Step 2: Find the constant \( k \)
We know that when \( x = 4 \), \( y = 2 \). Substitute these values into the equation to find \( k \):
\[
2 = \frac{k}{4^2}
\]
\[
2 = \frac{k}{16}
\]
Multiply both sides by 16:
\[
k = 32
\]

### Step 3: Rewrite the equation with the constant
Now that we know \( k = 32 \), the equation becomes:
\[
y = \frac{32}{x^2}
\]

### Step 4: Find \( y \) when \( x = \frac{1}{2} \)
Substitute \( x = \frac{1}{2} \) into the equation:
\[
y = \frac{32}{\left( \frac{1}{2} \right)^2}
\]
Calculate \( \left( \frac{1}{2} \right)^2 \):
\[
\left( \frac{1}{2} \right)^2 = \frac{1}{4}
\]
So the equation becomes:
\[
y = \frac{32}{\frac{1}{4}}
\]
Dividing by \( \frac{1}{4} \) is the same as multiplying by 4:
\[
y = 32 \times 4 = 128
\]

### Final Answer
\[
y = 128
\]

Would you like a further breakdown of these steps, or do you have any questions?

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Here are five related questions that might interest you:

1. How does inverse proportionality differ from direct proportionality?
2. What happens to \( y \) if \( x \) is doubled in this inverse relationship?
3. How can we interpret inverse proportionality graphically?
4. How would the relationship change if \( y \) were proportional to \( x \) instead of \( x^2 \)?
5. Can this approach be used for other powers of \( x \), such as \( x^3 \)?

**Tip:** Remember, when dealing with inverse proportionality, as \( x \) increases, \( y \) decreases, and vice versa.

y is inversely proportional to x^2. When x = 4, y = 2. Find y when x = 1/2.

Learn how to solve an inverse proportion problem where y is inversely proportional to x^2. Given that y = 2 when x = 4, find y when x = 1/2. This solution involves algebraic concepts and exponents, suitable for Grades 8-10.

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