Given that \( P \propto \frac{1}{\sqrt{r}} \), we can express this relationship as:

\(P = k \cdot \frac{1}{\sqrt{r}}\)

for some constant \( k \).

From the information provided, when \( P = 3 \) and \( r = 16 \):

\(3 = k \cdot \frac{1}{\sqrt{16}}\)

Since \( \sqrt{16} = 4 \), we have: \(3 = k \cdot \frac{1}{4}\)

Thus, \(k = 3 \cdot 4 = 12\)

Now we can write the equation as: \(P = \frac{12}{\sqrt{r}}\)

Next, we need to find \( r \) when \( P = \frac{3}{2} \):

\(\frac{3}{2} = \frac{12}{\sqrt{r}}\)

Cross-multiplying gives:

\(3\sqrt{r} = 24\)

Dividing both sides by 3:

\(\sqrt{r} = 8\)

Squaring both sides:

\(r = 64\). Thus, r = 64