To solve this problem, we can use the concept of joint variation, which states that \( C = k \cdot W \cdot T \), where:
- \( C \) is the number of cars produced,
- \( W \) is the number of workers,
- \( T \) is the time they work,
- \( k \) is the constant of variation.
We start by finding the value of \( k \) using the initial conditions given: 220 workers can produce 198 cars in 5 hours.
Plugging in the known values: \[ 198 = k \cdot 220 \cdot 5 \]
Now, solve for \( k \): \[ 198 = k \cdot 1100 \] \[ k = \frac{198}{1100} \] \[ k = \frac{99}{550} = \frac{9}{50} \]
Now that we have the constant of variation \( k \), we can use it to find the number of workers needed to produce 189 cars in 6 hours.
We set up the equation using \( k \): \[ 189 = \frac{9}{50} \cdot W \cdot 6 \]
Solving for \( W \): \[ 189 = \frac{9 \cdot 6}{50} \cdot W \] \[ 189 = \frac{54}{50} \cdot W \]
To clear the fraction, we multiply both sides by \( \frac{50}{54} \): \[ W = 189 \cdot \frac{50}{54} \]
Calculating \( W \): \[ W = \frac{189 \cdot 50}{54} \] \[ W = \frac{9450}{54} \] \[ W = \frac{1575}{9} = 175 \]
Thus, the number of workers required to produce 189 cars in 6 hours is \( \boxed{175} \).
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