The weight W kg of a metal bar varies jointly as its length L meters and the square of its diameter d meters. If w = 140 when d = 42/3 and L = 54, find d in terms of W and L.
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, \( d = \sqrt{ \dfrac{42W}{5L} } \).
There is an explanation video available below.
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