aleks.com/alekscgi/X/sli.exe/ho_u-lgNsidirSRv2K3inMNn3SP43gZqKs7v03QBBzBbUiOF_77CDLH66miXH9QJirD9GisQ9967hw3DITVgmacAHyt2aL16rxCPIuQM7r8-k18... Cowboy Channelt Google Docs Dashboard Google Drive e2020.geniussis.c... Course Map - Ima... Wyzant Rational Expressions Word problem on combined variation Suppose that the number of cars manufactured at an automobile plant varies jointly as the number of workers and the time they work. If 340 workers can produce 204 cars in 4 hours, find the number of hours required for 400 workers to produce 180 cars. \( \square \) hours

We are given that the number of cars produced varies jointly as the number of workers and the number of hours worked. This can be modeled as \[ C = k \cdot W \cdot t, \] where \( C \) is the number of cars produced, \( W \) is the number of workers, \( t \) is the time in hours, and \( k \) is the constant of proportionality. **Step 1. Solve for \( k \) using the initial conditions.** It is given that 340 workers produce 204 cars in 4 hours. Substitute these values into the equation: \[ 204 = k \cdot 340 \cdot 4. \] Solve for \( k \): \[ k = \frac{204}{340 \cdot 4}. \] Simplify the denominator: \[ 340 \cdot 4 = 1360, \] so \[ k = \frac{204}{1360}. \] Simplify the fraction by dividing the numerator and denominator by 68: \[ 204 \div 68 = 3 \quad \text{and} \quad 1360 \div 68 = 20, \] which gives \[ k = \frac{3}{20}. \] **Step 2. Use \( k \) to find the required time.** Now, for the second scenario, 400 workers are to produce 180 cars. Substitute \( W = 400 \), \( C = 180 \), and \( k = \frac{3}{20} \) into the equation: \[ 180 = \frac{3}{20} \cdot 400 \cdot t. \] Compute \( \frac{3}{20} \cdot 400 \): \[ \frac{3}{20} \cdot 400 = 3 \cdot 20 = 60. \] So the equation becomes: \[ 180 = 60 \cdot t. \] Solve for \( t \): \[ t = \frac{180}{60} = 3. \] Thus, the number of hours required is \[ \boxed{3}. \]