To solve the problem, we need to understand the relationship between the length \( L \) of a simple pendulum and its period \( T \). The problem states that the length varies directly as the square of the period. This can be expressed mathematically as: \[ L = k \cdot T^2 \] where \( k \) is a constant of proportionality. ### Step 1: Find the constant \( k \) We know from the problem that when the period \( T = 4 \) seconds, the length \( L = 64 \) cm. We can use this information to find the value of \( k \). Substituting the known values into the equation: \[ 64 = k \cdot (4^2) \] Calculating \( 4^2 \): \[ 4^2 = 16 \] Now substituting this back into the equation: \[ 64 = k \cdot 16 \] To find \( k \), we divide both sides by 16: \[ k = \frac{64}{16} = 4 \] ### Step 2: Write the equation for \( L \) Now that we have \( k \), we can write the equation for the length of the pendulum in terms of the period: \[ L = 4 \cdot T^2 \] ### Step 3: Find the length for \( T = 9 \) seconds Next, we need to find the length \( L \) when the period \( T = 9 \) seconds. We substitute \( T = 9 \) into our equation: \[ L = 4 \cdot (9^2) \] Calculating \( 9^2 \): \[ 9^2 = 81 \] Now substituting this back into the equation: \[ L = 4 \cdot 81 \] Calculating \( 4 \cdot 81 \): \[ L = 324 \text{ cm} \] ### Conclusion Thus, the length of the pendulum whose period is 9 seconds is **324 cm**. Therefore, the correct option is **B**. ### Explanation of Other Options - **Option A: 96 cm** - This value is too low. It does not take into account the square of the period correctly. - **Option C: 36 cm** - This value is also incorrect. It does not reflect the relationship between length and the square of the period. - **Option D: 144 cm** - This value is incorrect as well. It does not match the calculated length based on the given period. ### Revision Summary - The length \( L \) of a pendulum varies directly as the square of its period \( T \): \( L = k \cdot T^2 \). - To find the constant \( k \), use known values of \( L \) and \( T \). - Substitute the desired period into the equation to find the new length. - Always check the calculations to ensure accuracy and understand the relationships involved.