To solve the problem where \( x \) varies directly as \( \sqrt{n} \), we start by understanding the concept of direct variation. When we say that \( x \) varies directly as \( \sqrt{n} \), it means that there exists a constant \( k \) such that: \[ x = k \cdot \sqrt{n} \] ### Step 1: Find the constant \( k \) We are given that \( x = 9 \) when \( n = 9 \). We can use this information to find the value of \( k \). Substituting the known values into the equation: \[ 9 = k \cdot \sqrt{9} \] Since \( \sqrt{9} = 3 \), we can rewrite the equation as: \[ 9 = k \cdot 3 \] To isolate \( k \), we divide both sides by 3: \[ k = \frac{9}{3} = 3 \] ### Step 2: Write the equation for \( x \) Now that we have found \( k \), we can write the equation that relates \( x \) and \( n \): \[ x = 3 \cdot \sqrt{n} \] ### Step 3: Find \( x \) when \( n = \frac{17}{9} \) Next, we need to find \( x \) when \( n = \frac{17}{9} \). We substitute \( n \) into our equation: \[ x = 3 \cdot \sqrt{\frac{17}{9}} \] ### Step 4: Simplify \( \sqrt{\frac{17}{9}} \) To simplify \( \sqrt{\frac{17}{9}} \), we can separate the square root: \[ \sqrt{\frac{17}{9}} = \frac{\sqrt{17}}{\sqrt{9}} = \frac{\sqrt{17}}{3} \] ### Step 5: Substitute back into the equation for \( x \) Now we substitute this back into our equation for \( x \): \[ x = 3 \cdot \frac{\sqrt{17}}{3} \] The \( 3 \) in the numerator and denominator cancels out: \[ x = \sqrt{17} \] ### Conclusion Thus, the value of \( x \) when \( n = \frac{17}{9} \) is: \[ \boxed{\sqrt{17}} \] ### Explanation of Other Options - **Option A: 4** - This is incorrect because it does not satisfy the relationship derived from the direct variation equation. - **Option B: 27** - This is also incorrect as it does not align with the calculated value of \( x \) based on the given \( n \). - **Option C: \( \sqrt{3} \)** - This option is incorrect as it does not correspond to the derived value of \( x \) when substituting \( n = \frac{17}{9} \). ### Revision Summary - Direct variation means \( x = k \cdot \sqrt{n} \) for some constant \( k \). - To find \( k \), use known values of \( x \) and \( n \). - Substitute the new value of \( n \) into the equation to find \( x \). - Simplify square roots carefully to avoid mistakes in calculations.