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