\( y \) is inversely proportional to the square root of \( x \), and \( y=9 \) when \( x=16 \). Find \( y \) when \( x=8 \)

Since \( y \) is inversely proportional to the square root of \( x \), we can express this relationship mathematically as: \[ y = \frac{k}{\sqrt{x}} \] where \( k \) is a constant. Given that \( y = 9 \) when \( x = 16 \), we can substitute these values into the equation to find \( k \): \[ 9 = \frac{k}{\sqrt{16}} \] Since \( \sqrt{16} = 4 \), we can rewrite the equation as: \[ 9 = \frac{k}{4} \] To find \( k \), we multiply both sides by 4: \[ k = 9 \times 4 = 36 \] Now that we have the value of \( k \), we can write the equation for \( y \): \[ y = \frac{36}{\sqrt{x}} \] Next, we need to find \( y \) when \( x = 8 \): \[ y = \frac{36}{\sqrt{8}} \] Calculating \( \sqrt{8} \): \[ \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \] Now substituting this back into the equation for \( y \): \[ y = \frac{36}{2\sqrt{2}} = \frac{36}{2} \cdot \frac{1}{\sqrt{2}} = 18 \cdot \frac{1}{\sqrt{2}} = \frac{18}{\sqrt{2}} \] To rationalize the denominator, we multiply the numerator and the denominator by \( \sqrt{2} \): \[ y = \frac{18\sqrt{2}}{2} = 9\sqrt{2} \] Thus, the value of \( y \) when \( x = 8 \) is: \[ \boxed{9\sqrt{2}} \]