To solve the problem, we need to understand the relationship given in the question. The weight \( W \) of a metal bar varies jointly as its length \( L \) and the square of its diameter \( d \). This can be expressed mathematically as: \[ W = k \cdot L \cdot d^2 \] where \( k \) is a constant of proportionality. ### Step 1: Find the Constant \( k \) We are given that \( W = 140 \) when \( d = 4\frac{2}{3} \) and \( L = 54 \). First, we need to convert \( d \) into an improper fraction or a decimal for easier calculations. The mixed number \( 4\frac{2}{3} \) can be converted as follows: \[ 4\frac{2}{3} = 4 + \frac{2}{3} = \frac{12}{3} + \frac{2}{3} = \frac{14}{3} \] Now we can substitute \( W \), \( L \), and \( d \) into the equation to find \( k \): \[ 140 = k \cdot 54 \cdot \left(\frac{14}{3}\right)^2 \] Calculating \( \left(\frac{14}{3}\right)^2 \): \[ \left(\frac{14}{3}\right)^2 = \frac{196}{9} \] Now substituting this back into the equation: \[ 140 = k \cdot 54 \cdot \frac{196}{9} \] To isolate \( k \), we rearrange the equation: \[ k = \frac{140 \cdot 9}{54 \cdot 196} \] Calculating the right side: 1. Calculate \( 140 \cdot 9 = 1260 \). 2. Calculate \( 54 \cdot 196 = 10584 \). Now we have: \[ k = \frac{1260}{10584} \] To simplify this fraction, we can divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 1260 and 10584 is 126. Thus: \[ k = \frac{10}{84} = \frac{5}{42} \] ### Step 2: Substitute \( k \) Back into the Original Equation Now that we have \( k \), we can substitute it back into the original equation: \[ W = \frac{5}{42} \cdot L \cdot d^2 \] ### Step 3: Solve for \( d \) We want to express \( d \) in terms of \( W \) and \( L \). Rearranging the equation gives: \[ d^2 = \frac{42W}{5L} \] Taking the square root of both sides to solve for \( d \): \[ d = \sqrt{\frac{42W}{5L}} \] ### Conclusion: Identify the Correct Option Now, let's compare our derived formula for \( d \) with the provided options: - **Option A:** \( d = \sqrt{\frac{42W}{5L}} \) (Correct) - **Option B:** \( d = \sqrt{\frac{6L}{42W}} \) (Incorrect) - **Option C:** \( d = \frac{42W}{5L} \) (Incorrect) - **Option D:** \( d = \frac{5L}{42W} \) (Incorrect) ### Explanation of Incorrect Options - **Option B:** This option incorrectly rearranges the terms and does not maintain the correct relationship between \( W \), \( L \), and \( d \). - **Option C:** This option suggests a direct proportionality rather than a square relationship, which is not consistent with the original equation. - **Option D:** This option also misrepresents the relationship, suggesting an inverse relationship rather than the correct joint variation. ### Revision Summary - The weight \( W \) varies jointly as the length \( L \) and the square of the diameter \( d \). - The formula derived is \( d = \sqrt{\frac{42W}{5L}} \). - The constant \( k \) was found to be \( \frac{5}{42} \) using the given values. - Always check the relationships and units when dealing with joint variation problems.