# Constraints Production Capacity Let- \( X_1 \) = number of tables - \( X_2 \) = number of chairs ## Carpentry Constraint - Each table requires 4 hours - Each chair requires 3 hours - Total carpentry hours available = 1,200 **Equation:** \[ 4X_1 + 3X_2 \leq 120 \] ## Polishing Constraint - Each table requires 2 hours - Each chair requires 1 hour - Total polishing hours available = 500 **Equation:** \[ 2X_1 + 1X_2 \leq 500 \] --- # Production Quantities if Only Carpentry is Fully Used Set the carpentry constraint to equality: \[ 4X_1 + 3X_2 = 120 \] Assuming the polishing constraint is not limiting (i.e., \( 2X_1 + X_2 \geq \) some value), we focus on maximizing \( X_1, X_2 \) within carpentry hours. ### To find the maximum production: - **Maximize \( X_1 \) when \( X_2 = \):** \[ 4X_1 = 120 \Rightarrow X_1 = \frac{120}{4} = 300 \] - **Maximize \( X_2 \) when \( X_1 = \):** \[ 3X_2 = 120 \Rightarrow X_2 = \frac{120}{3} = 400 \] ### To produce both simultaneously: - Find the combination where \( 4X_1 + 3X_2 = 120 \) and \( 2X_1 + X_2 \) is within polishing hours. - For the maximum number of tables and chairs respecting only carpentry: \[ \boxed{ \text{Maximum tables} = 300 \quad \text{(if no chairs are produced)} } \] \[ \boxed{ \text{Maximum chairs} = 400 \quad \text{(if no tables are produced)} } \] Any combination along the line \( 4X_1 + 3X_2 = 120 \) that respects the polishing constraint can be considered, but these represent the extremes.