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?

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.