The weight \( W \) kg of a metal bar varies jointly as its length \( L \) meters and the square of its diameter \( d \) meters. This relationship is expressed as: \(W = k \cdot L \cdot d^2\) where \( k \) is the constant of proportionality. Given \( W = 140 \), \( d = 4\frac{2}{3} = \frac{14}{3} \), and \( L = 54 \), substitute to find \( k \): \(140 = k \cdot 54 \cdot \left( \frac{14}{3} \right)^2\) Calculate \( \left( \frac{14}{3} \right)^2 = \frac{196}{9} \): \(140 = k \cdot 54 \cdot \frac{196}{9}\) Simplify: \( 54 \div 9 = 6 \), so: \(140 = k \cdot 6 \cdot 196 = k \cdot 1176\) Thus, \(k = \frac{140}{1176} = \frac{5}{42}\) Now rearrange the original equation to solve for \( d \): \(d^2 = \frac{W}{k \cdot L}\) \(d = \sqrt{ \frac{W}{k \cdot L} } = \sqrt{ \frac{W}{\frac{5}{42} \cdot L} } = \sqrt{ \frac{42W}{5L} }\) Therefore, \( d = \sqrt{ \dfrac{42W}{5L} } \).